Page 1 DIFFERENTIAL EQUATIONS: AN INTRODUCTION MATH 15300, SECTION 21 (VIPUL NAIK) Corresponding material in the book: Section 9.2. What students should already know: The prime and Leibniz notation for derivatives, the meaning of dierentiation, implicit dierentiation, and integration. What students should denitely get: What a dierential equation means, what a solution to a dierential equation means, how to solve a multiplicatively separable rst-order dierential equation, how to solve an initial value problem. What students should hopefully get: The notion of parameters as degrees of freedom, the notion of constraints as pinning these down, the basic concerns in dierential equation manipulation. 1. Understanding differential equations and solutions 1.1. Dierentiation: the two interpretations. We have dealt with two interpretations of dierentiation that it would be useful to recall at this stage. One interpretation is in terms of functions. Here, we think of a functionf as a black box that takes as input a variablex and outputs a variablef(x), that we may choose to call y. f 0 is a new function, i.e., a new black box, that takes as input x and gives an output called f 0 (x), that we may also call y 0 . In this interpretation, it is the function, rather than the inputs and outputs to it, that takes on primal importance. The disadvantage of this approach is that it does not allow us to go beyond functions. The second interpretation is to view a function as a relation between two quantities { the input quantity and the output quantity. The function describes the nature of the dependence of the output quantity upon the input quantity. Under this approach, we denote the derivative as dy=dx, the Leibniz notation. The Leibniz notation dy=dx arises from the fact that the derivative is the limit of the dierence quotient: dy dx = lim y x The focus here is not on the function that relates x to y, but on the variables x and y. The advantage of this approach is that we can apply this approach even when neither of the variables is a function of the other. For instance, we could do something called implicit dierentiation, which allows us to nd dy=dx when x and y are entangled. For instance, given: y 2 + sin(xy) =x 3 cos(x +y) We dierentiate and get: 2y dy dx + cos(xy) x dy dx +y = 3x 2 cos(x +y)x 3 sin(x +y) 1 + dy dx We can collect terms and obtain an expression for dy=dx in terms of x and y. 1.2. A dierential equation. Consider two variables x (the so-called independent variable) and y (the so-called dependent variable). A dierential equation is an equation involving the variables x, y, and rst and higher derivatives of y with respect to x. For instance, here's a dierential equation. x +yy 0 +xy sin(y 0 ) = 0 Here, y 0 is shorthand for dy=dx. Thus, this dierential equation can also be written as: x +y dy dx +xy sin dy dx = 0 1 Page 2 DIFFERENTIAL EQUATIONS: AN INTRODUCTION MATH 15300, SECTION 21 (VIPUL NAIK) Corresponding material in the book: Section 9.2. What students should already know: The prime and Leibniz notation for derivatives, the meaning of dierentiation, implicit dierentiation, and integration. What students should denitely get: What a dierential equation means, what a solution to a dierential equation means, how to solve a multiplicatively separable rst-order dierential equation, how to solve an initial value problem. What students should hopefully get: The notion of parameters as degrees of freedom, the notion of constraints as pinning these down, the basic concerns in dierential equation manipulation. 1. Understanding differential equations and solutions 1.1. Dierentiation: the two interpretations. We have dealt with two interpretations of dierentiation that it would be useful to recall at this stage. One interpretation is in terms of functions. Here, we think of a functionf as a black box that takes as input a variablex and outputs a variablef(x), that we may choose to call y. f 0 is a new function, i.e., a new black box, that takes as input x and gives an output called f 0 (x), that we may also call y 0 . In this interpretation, it is the function, rather than the inputs and outputs to it, that takes on primal importance. The disadvantage of this approach is that it does not allow us to go beyond functions. The second interpretation is to view a function as a relation between two quantities { the input quantity and the output quantity. The function describes the nature of the dependence of the output quantity upon the input quantity. Under this approach, we denote the derivative as dy=dx, the Leibniz notation. The Leibniz notation dy=dx arises from the fact that the derivative is the limit of the dierence quotient: dy dx = lim y x The focus here is not on the function that relates x to y, but on the variables x and y. The advantage of this approach is that we can apply this approach even when neither of the variables is a function of the other. For instance, we could do something called implicit dierentiation, which allows us to nd dy=dx when x and y are entangled. For instance, given: y 2 + sin(xy) =x 3 cos(x +y) We dierentiate and get: 2y dy dx + cos(xy) x dy dx +y = 3x 2 cos(x +y)x 3 sin(x +y) 1 + dy dx We can collect terms and obtain an expression for dy=dx in terms of x and y. 1.2. A dierential equation. Consider two variables x (the so-called independent variable) and y (the so-called dependent variable). A dierential equation is an equation involving the variables x, y, and rst and higher derivatives of y with respect to x. For instance, here's a dierential equation. x +yy 0 +xy sin(y 0 ) = 0 Here, y 0 is shorthand for dy=dx. Thus, this dierential equation can also be written as: x +y dy dx +xy sin dy dx = 0 1 If we want to get y =f(x), the above can be rewritten as: x +f(x)f 0 (x) +xf(x) sin(f 0 (x)) = 0 Another way of putting this is that a dierential equation is something of the form F (x;y;y 0 ;y 00 ;::: ) = 0 where F is some expression in many variables. Before proceeding further, however, we must understand what a dierential equation means, and how it diers from an ordinary equation. (1) A functional solution or function solution is a function y = f(x) such that, taking derivatives the usual way, we nd that the dierential equation is satised for all x. More specically, a function y =f(x) solves the dierential equation if F (x;f(x);f 0 (x);f 00 (x);::: ) = 0 for all x. (2) A solution to a dierential equation is a relation R(x;y) between x and y such that if we consider the set R(x;y) = 0, and use implicit dierentiation to nd the higher derivatives, these satisfy the condition F 0. A solution may dier from a functional solution in the sense that y may not be written explicitly as a function of x. (3) More pictorially, a solution to a dierential equation is a curve in the planeR 2 with the property that the dierential equation holds at all points on the curve. What this means is that at any point on the curve, if we calculate the higher derivatives based on their geometric interpretations, we obtain a bunch of stu that satises the dierential equation. What does this mean and how does this dier from an ordinary equation? Two important dierences: (1) Each solution to an ordinary equation (such as a polynomial equation) is a number. In contrast, each solution to a dierential equation is a function or relation between two variables. (2) When we are looking at an ordinary equation, such asx 2 +x + 2 = 0, we are looking at points where this equation holds. When we are looking at a dierential equation, we are looking at curves such that the equation holds at all points on the curve. (3) To check that an ordinary equation holds at a point, we evaluate at the point. However, to check whether a dierential equation holds, we need to understand behavior locally, on a neighborhood. In other words, it makes no sense to ask whether a dierential equation holds for a given point (x 0 ;y 0 ); it only makes sense to ask whether it holds on a given curve. Basically, a dierential equation seeks to nd a function that exhibits certain local behavior as described by an expression involving the function and its derivatives. Aside: Dierential equations as functional equations. A functional equation is an equation that asks for a function satisfying certain conditions. Specically, a functional equation is an equation in terms of a function that we require to be true for every choice of value for all the letters in the equation, i.e., we require it to be an identity in all letter variables. 1 For instance, the equation: f(x) =f(x)8 x2R has solution set precisely the set of all even functions. Similarly, the equation: f(ax) =af(x)8a;x2R has solution set precisely the set of all functions f of the form f(x) =x, where is a constant. The desired solutions to functional equations are functions, and is does not make sense to ask whether a particular input-output pair satises a functional equation. Dierential equations are a particular kind of functional equations. Specically, dierential equations are functional equations involving derivative behavior all considered at a single point. 2 1 In mathematical jargon, the letter variables for numbers are typically quantied over all integers. 2 There are more complicated functional equations involving derivatives that are not dierential equations in the sense that we have talked about. Examples include delay dierential equations. 2 Page 3 DIFFERENTIAL EQUATIONS: AN INTRODUCTION MATH 15300, SECTION 21 (VIPUL NAIK) Corresponding material in the book: Section 9.2. What students should already know: The prime and Leibniz notation for derivatives, the meaning of dierentiation, implicit dierentiation, and integration. What students should denitely get: What a dierential equation means, what a solution to a dierential equation means, how to solve a multiplicatively separable rst-order dierential equation, how to solve an initial value problem. What students should hopefully get: The notion of parameters as degrees of freedom, the notion of constraints as pinning these down, the basic concerns in dierential equation manipulation. 1. Understanding differential equations and solutions 1.1. Dierentiation: the two interpretations. We have dealt with two interpretations of dierentiation that it would be useful to recall at this stage. One interpretation is in terms of functions. Here, we think of a functionf as a black box that takes as input a variablex and outputs a variablef(x), that we may choose to call y. f 0 is a new function, i.e., a new black box, that takes as input x and gives an output called f 0 (x), that we may also call y 0 . In this interpretation, it is the function, rather than the inputs and outputs to it, that takes on primal importance. The disadvantage of this approach is that it does not allow us to go beyond functions. The second interpretation is to view a function as a relation between two quantities { the input quantity and the output quantity. The function describes the nature of the dependence of the output quantity upon the input quantity. Under this approach, we denote the derivative as dy=dx, the Leibniz notation. The Leibniz notation dy=dx arises from the fact that the derivative is the limit of the dierence quotient: dy dx = lim y x The focus here is not on the function that relates x to y, but on the variables x and y. The advantage of this approach is that we can apply this approach even when neither of the variables is a function of the other. For instance, we could do something called implicit dierentiation, which allows us to nd dy=dx when x and y are entangled. For instance, given: y 2 + sin(xy) =x 3 cos(x +y) We dierentiate and get: 2y dy dx + cos(xy) x dy dx +y = 3x 2 cos(x +y)x 3 sin(x +y) 1 + dy dx We can collect terms and obtain an expression for dy=dx in terms of x and y. 1.2. A dierential equation. Consider two variables x (the so-called independent variable) and y (the so-called dependent variable). A dierential equation is an equation involving the variables x, y, and rst and higher derivatives of y with respect to x. For instance, here's a dierential equation. x +yy 0 +xy sin(y 0 ) = 0 Here, y 0 is shorthand for dy=dx. Thus, this dierential equation can also be written as: x +y dy dx +xy sin dy dx = 0 1 If we want to get y =f(x), the above can be rewritten as: x +f(x)f 0 (x) +xf(x) sin(f 0 (x)) = 0 Another way of putting this is that a dierential equation is something of the form F (x;y;y 0 ;y 00 ;::: ) = 0 where F is some expression in many variables. Before proceeding further, however, we must understand what a dierential equation means, and how it diers from an ordinary equation. (1) A functional solution or function solution is a function y = f(x) such that, taking derivatives the usual way, we nd that the dierential equation is satised for all x. More specically, a function y =f(x) solves the dierential equation if F (x;f(x);f 0 (x);f 00 (x);::: ) = 0 for all x. (2) A solution to a dierential equation is a relation R(x;y) between x and y such that if we consider the set R(x;y) = 0, and use implicit dierentiation to nd the higher derivatives, these satisfy the condition F 0. A solution may dier from a functional solution in the sense that y may not be written explicitly as a function of x. (3) More pictorially, a solution to a dierential equation is a curve in the planeR 2 with the property that the dierential equation holds at all points on the curve. What this means is that at any point on the curve, if we calculate the higher derivatives based on their geometric interpretations, we obtain a bunch of stu that satises the dierential equation. What does this mean and how does this dier from an ordinary equation? Two important dierences: (1) Each solution to an ordinary equation (such as a polynomial equation) is a number. In contrast, each solution to a dierential equation is a function or relation between two variables. (2) When we are looking at an ordinary equation, such asx 2 +x + 2 = 0, we are looking at points where this equation holds. When we are looking at a dierential equation, we are looking at curves such that the equation holds at all points on the curve. (3) To check that an ordinary equation holds at a point, we evaluate at the point. However, to check whether a dierential equation holds, we need to understand behavior locally, on a neighborhood. In other words, it makes no sense to ask whether a dierential equation holds for a given point (x 0 ;y 0 ); it only makes sense to ask whether it holds on a given curve. Basically, a dierential equation seeks to nd a function that exhibits certain local behavior as described by an expression involving the function and its derivatives. Aside: Dierential equations as functional equations. A functional equation is an equation that asks for a function satisfying certain conditions. Specically, a functional equation is an equation in terms of a function that we require to be true for every choice of value for all the letters in the equation, i.e., we require it to be an identity in all letter variables. 1 For instance, the equation: f(x) =f(x)8 x2R has solution set precisely the set of all even functions. Similarly, the equation: f(ax) =af(x)8a;x2R has solution set precisely the set of all functions f of the form f(x) =x, where is a constant. The desired solutions to functional equations are functions, and is does not make sense to ask whether a particular input-output pair satises a functional equation. Dierential equations are a particular kind of functional equations. Specically, dierential equations are functional equations involving derivative behavior all considered at a single point. 2 1 In mathematical jargon, the letter variables for numbers are typically quantied over all integers. 2 There are more complicated functional equations involving derivatives that are not dierential equations in the sense that we have talked about. Examples include delay dierential equations. 2 1.3. Some examples of dierential equations and solutions. We begin with a simple dierential equation: dy dx = 1 We claim that y = x + 13 is a solution to this dierential equation. In the various jargon that we have introduced: (1) The function f(x) =x + 13 satises the condition that f 0 (x) = 1. (2) The curve y =x + 13 satises the condition y 0 = 1. Both these statements are clearly true. Graphically, the curve y =x + 13 is a straight line with slope 1 and intercept 13. Since its slope is 1, dy=dx = 1 everywhere on the line. However, this is not the only solution. Astute observers would have noted that there was nothing partic- ularly auspicious about the number 13. In fact, for any constantC,y =x+C, or the functionf(x) =x+C, solves this dierential equation. Thus, any line with slope 1 is a solution curve to this dierential equation. Note that every value ofC gives one solution, called a particular solution. Each such solution corresponds to a line with slope 1. Pictorially, we get a bunch of parallel lines that cover the entire plane. Are these the only ones? Indeed, which brings us to the next topic. 1.4. De ja vu. When we rst learn algebra in middle or high school, we are given examples such as: How many more apples need to be added to 3 apples to obtain 5 apples? Algebra version: Solve 3 +t = 5. At rst, these seem like silly examples, because anybody who has grasped the concept of subtraction (the inverse operation to addition) can probably solve the problem without any knowledge of algebra. It is only after seeing harder examples of equations that people begin to appreciate the power of algebraic manipulation in solving problems that are too hard to manipulate with basic arithmetic. In the same way, our rst example of a dierential equation is in fact an integration problem. Specically, solving: dy dx = 1 is equivalent to performing the indenite integration: y = Z 1dx which gives the answer y = x +C where C is an arbitrary constant. Each value of C gives a particular solution. Let us elaborate the process a little further: dy dx = 1 Moving the dx to the numerator on the other side, we obtain: dy =dx Note that this is just formal manipulation, akin to the way you learned formal algebra when you got started. Don't ponder about the intrinsic meaning of dy and dx. Integrating both sides, we get: Z dy = Z dx Which gives: y =x +C 3 Page 4 DIFFERENTIAL EQUATIONS: AN INTRODUCTION MATH 15300, SECTION 21 (VIPUL NAIK) Corresponding material in the book: Section 9.2. What students should already know: The prime and Leibniz notation for derivatives, the meaning of dierentiation, implicit dierentiation, and integration. What students should denitely get: What a dierential equation means, what a solution to a dierential equation means, how to solve a multiplicatively separable rst-order dierential equation, how to solve an initial value problem. What students should hopefully get: The notion of parameters as degrees of freedom, the notion of constraints as pinning these down, the basic concerns in dierential equation manipulation. 1. Understanding differential equations and solutions 1.1. Dierentiation: the two interpretations. We have dealt with two interpretations of dierentiation that it would be useful to recall at this stage. One interpretation is in terms of functions. Here, we think of a functionf as a black box that takes as input a variablex and outputs a variablef(x), that we may choose to call y. f 0 is a new function, i.e., a new black box, that takes as input x and gives an output called f 0 (x), that we may also call y 0 . In this interpretation, it is the function, rather than the inputs and outputs to it, that takes on primal importance. The disadvantage of this approach is that it does not allow us to go beyond functions. The second interpretation is to view a function as a relation between two quantities { the input quantity and the output quantity. The function describes the nature of the dependence of the output quantity upon the input quantity. Under this approach, we denote the derivative as dy=dx, the Leibniz notation. The Leibniz notation dy=dx arises from the fact that the derivative is the limit of the dierence quotient: dy dx = lim y x The focus here is not on the function that relates x to y, but on the variables x and y. The advantage of this approach is that we can apply this approach even when neither of the variables is a function of the other. For instance, we could do something called implicit dierentiation, which allows us to nd dy=dx when x and y are entangled. For instance, given: y 2 + sin(xy) =x 3 cos(x +y) We dierentiate and get: 2y dy dx + cos(xy) x dy dx +y = 3x 2 cos(x +y)x 3 sin(x +y) 1 + dy dx We can collect terms and obtain an expression for dy=dx in terms of x and y. 1.2. A dierential equation. Consider two variables x (the so-called independent variable) and y (the so-called dependent variable). A dierential equation is an equation involving the variables x, y, and rst and higher derivatives of y with respect to x. For instance, here's a dierential equation. x +yy 0 +xy sin(y 0 ) = 0 Here, y 0 is shorthand for dy=dx. Thus, this dierential equation can also be written as: x +y dy dx +xy sin dy dx = 0 1 If we want to get y =f(x), the above can be rewritten as: x +f(x)f 0 (x) +xf(x) sin(f 0 (x)) = 0 Another way of putting this is that a dierential equation is something of the form F (x;y;y 0 ;y 00 ;::: ) = 0 where F is some expression in many variables. Before proceeding further, however, we must understand what a dierential equation means, and how it diers from an ordinary equation. (1) A functional solution or function solution is a function y = f(x) such that, taking derivatives the usual way, we nd that the dierential equation is satised for all x. More specically, a function y =f(x) solves the dierential equation if F (x;f(x);f 0 (x);f 00 (x);::: ) = 0 for all x. (2) A solution to a dierential equation is a relation R(x;y) between x and y such that if we consider the set R(x;y) = 0, and use implicit dierentiation to nd the higher derivatives, these satisfy the condition F 0. A solution may dier from a functional solution in the sense that y may not be written explicitly as a function of x. (3) More pictorially, a solution to a dierential equation is a curve in the planeR 2 with the property that the dierential equation holds at all points on the curve. What this means is that at any point on the curve, if we calculate the higher derivatives based on their geometric interpretations, we obtain a bunch of stu that satises the dierential equation. What does this mean and how does this dier from an ordinary equation? Two important dierences: (1) Each solution to an ordinary equation (such as a polynomial equation) is a number. In contrast, each solution to a dierential equation is a function or relation between two variables. (2) When we are looking at an ordinary equation, such asx 2 +x + 2 = 0, we are looking at points where this equation holds. When we are looking at a dierential equation, we are looking at curves such that the equation holds at all points on the curve. (3) To check that an ordinary equation holds at a point, we evaluate at the point. However, to check whether a dierential equation holds, we need to understand behavior locally, on a neighborhood. In other words, it makes no sense to ask whether a dierential equation holds for a given point (x 0 ;y 0 ); it only makes sense to ask whether it holds on a given curve. Basically, a dierential equation seeks to nd a function that exhibits certain local behavior as described by an expression involving the function and its derivatives. Aside: Dierential equations as functional equations. A functional equation is an equation that asks for a function satisfying certain conditions. Specically, a functional equation is an equation in terms of a function that we require to be true for every choice of value for all the letters in the equation, i.e., we require it to be an identity in all letter variables. 1 For instance, the equation: f(x) =f(x)8 x2R has solution set precisely the set of all even functions. Similarly, the equation: f(ax) =af(x)8a;x2R has solution set precisely the set of all functions f of the form f(x) =x, where is a constant. The desired solutions to functional equations are functions, and is does not make sense to ask whether a particular input-output pair satises a functional equation. Dierential equations are a particular kind of functional equations. Specically, dierential equations are functional equations involving derivative behavior all considered at a single point. 2 1 In mathematical jargon, the letter variables for numbers are typically quantied over all integers. 2 There are more complicated functional equations involving derivatives that are not dierential equations in the sense that we have talked about. Examples include delay dierential equations. 2 1.3. Some examples of dierential equations and solutions. We begin with a simple dierential equation: dy dx = 1 We claim that y = x + 13 is a solution to this dierential equation. In the various jargon that we have introduced: (1) The function f(x) =x + 13 satises the condition that f 0 (x) = 1. (2) The curve y =x + 13 satises the condition y 0 = 1. Both these statements are clearly true. Graphically, the curve y =x + 13 is a straight line with slope 1 and intercept 13. Since its slope is 1, dy=dx = 1 everywhere on the line. However, this is not the only solution. Astute observers would have noted that there was nothing partic- ularly auspicious about the number 13. In fact, for any constantC,y =x+C, or the functionf(x) =x+C, solves this dierential equation. Thus, any line with slope 1 is a solution curve to this dierential equation. Note that every value ofC gives one solution, called a particular solution. Each such solution corresponds to a line with slope 1. Pictorially, we get a bunch of parallel lines that cover the entire plane. Are these the only ones? Indeed, which brings us to the next topic. 1.4. De ja vu. When we rst learn algebra in middle or high school, we are given examples such as: How many more apples need to be added to 3 apples to obtain 5 apples? Algebra version: Solve 3 +t = 5. At rst, these seem like silly examples, because anybody who has grasped the concept of subtraction (the inverse operation to addition) can probably solve the problem without any knowledge of algebra. It is only after seeing harder examples of equations that people begin to appreciate the power of algebraic manipulation in solving problems that are too hard to manipulate with basic arithmetic. In the same way, our rst example of a dierential equation is in fact an integration problem. Specically, solving: dy dx = 1 is equivalent to performing the indenite integration: y = Z 1dx which gives the answer y = x +C where C is an arbitrary constant. Each value of C gives a particular solution. Let us elaborate the process a little further: dy dx = 1 Moving the dx to the numerator on the other side, we obtain: dy =dx Note that this is just formal manipulation, akin to the way you learned formal algebra when you got started. Don't ponder about the intrinsic meaning of dy and dx. Integrating both sides, we get: Z dy = Z dx Which gives: y =x +C 3 (Note that we can actually get positive constants with both indenite integrals, but we can absorb the two constants into one). This drawn-out process seems pointless for this specic example, just like algebra seemed pointless when you were solving 3 +t = 5. Unlike middle school, however, where we waded through a lot of these silly examples before getting to more substantive examples of the use of algebra, we can now jump straight to harder situations where we see the machinery of dierential equations and how it gets used. Aside: Yesterday's problem is today's solution. In the elementary grades, addition, subtraction, multiplication and division were the problems, and numbers were the answers. In the middle grades, simple algebraic equations were the problems, and reducing them to a clearly stated arithmetic computation was the crux of the solution (the rest was trivial). When we learned dierentiation, dierentiating functions was the problem, but when we studied graphing and integration, dierentiation was one of the many tools that was used in coming up with solutions. This is the fundamental nature of mathematics: yesterday's focus problems become the taken-for-granted tools of solution to today's focus problems. Reducing the solution to today's focus problem to solving a bunch of yesterday's focus problems is almost as good at solving yesterday's problem. 3 Until very recently in this course, integration was the problem. Now, integration is part of the solution. Once a dierential equation is reduced to calculating a bunch of integrals, we're home. Our goal in nding solution functions to dierential equations is to reduce dierential equations to (one or more) integration problems. 2. Understanding and solving separable equations 2.1. Separable equations: a crash course. A dierential equation of the form: dy dx =f(x) has as solution: y = Z f(x)dx where the right side can be computed by nding one antiderivative and then tacking on a +C. This is a form of dierential equation that directly reduces to a single integration problem { the calculus analogue to 3 +t = 5. Let's look at a slightly more complicated example: dy dx =f(x)g(y) also written as: y 0 =f(x)g(y) In words, we say thaty 0 is a multiplicatively separable function ofx andy { it is the product of a function that depends only on x and a function that depends only on y. We do some algebra-like manipulations whose aim is to bring together on one side all terms involving y and bring together on the other side all terms involving x: dy g(y) =f(x)dx We now put integral signs and carry out indenite integration: Z dy g(y) = Z f(x)dx 3 Assuming you remember how to solve yesterday's problems. Mathematical knowledge is cumulative, but an individual's mathematical knowledge is cumulative only if that individual actually accumulates knowledge. 4 Page 5 DIFFERENTIAL EQUATIONS: AN INTRODUCTION MATH 15300, SECTION 21 (VIPUL NAIK) Corresponding material in the book: Section 9.2. What students should already know: The prime and Leibniz notation for derivatives, the meaning of dierentiation, implicit dierentiation, and integration. What students should denitely get: What a dierential equation means, what a solution to a dierential equation means, how to solve a multiplicatively separable rst-order dierential equation, how to solve an initial value problem. What students should hopefully get: The notion of parameters as degrees of freedom, the notion of constraints as pinning these down, the basic concerns in dierential equation manipulation. 1. Understanding differential equations and solutions 1.1. Dierentiation: the two interpretations. We have dealt with two interpretations of dierentiation that it would be useful to recall at this stage. One interpretation is in terms of functions. Here, we think of a functionf as a black box that takes as input a variablex and outputs a variablef(x), that we may choose to call y. f 0 is a new function, i.e., a new black box, that takes as input x and gives an output called f 0 (x), that we may also call y 0 . In this interpretation, it is the function, rather than the inputs and outputs to it, that takes on primal importance. The disadvantage of this approach is that it does not allow us to go beyond functions. The second interpretation is to view a function as a relation between two quantities { the input quantity and the output quantity. The function describes the nature of the dependence of the output quantity upon the input quantity. Under this approach, we denote the derivative as dy=dx, the Leibniz notation. The Leibniz notation dy=dx arises from the fact that the derivative is the limit of the dierence quotient: dy dx = lim y x The focus here is not on the function that relates x to y, but on the variables x and y. The advantage of this approach is that we can apply this approach even when neither of the variables is a function of the other. For instance, we could do something called implicit dierentiation, which allows us to nd dy=dx when x and y are entangled. For instance, given: y 2 + sin(xy) =x 3 cos(x +y) We dierentiate and get: 2y dy dx + cos(xy) x dy dx +y = 3x 2 cos(x +y)x 3 sin(x +y) 1 + dy dx We can collect terms and obtain an expression for dy=dx in terms of x and y. 1.2. A dierential equation. Consider two variables x (the so-called independent variable) and y (the so-called dependent variable). A dierential equation is an equation involving the variables x, y, and rst and higher derivatives of y with respect to x. For instance, here's a dierential equation. x +yy 0 +xy sin(y 0 ) = 0 Here, y 0 is shorthand for dy=dx. Thus, this dierential equation can also be written as: x +y dy dx +xy sin dy dx = 0 1 If we want to get y =f(x), the above can be rewritten as: x +f(x)f 0 (x) +xf(x) sin(f 0 (x)) = 0 Another way of putting this is that a dierential equation is something of the form F (x;y;y 0 ;y 00 ;::: ) = 0 where F is some expression in many variables. Before proceeding further, however, we must understand what a dierential equation means, and how it diers from an ordinary equation. (1) A functional solution or function solution is a function y = f(x) such that, taking derivatives the usual way, we nd that the dierential equation is satised for all x. More specically, a function y =f(x) solves the dierential equation if F (x;f(x);f 0 (x);f 00 (x);::: ) = 0 for all x. (2) A solution to a dierential equation is a relation R(x;y) between x and y such that if we consider the set R(x;y) = 0, and use implicit dierentiation to nd the higher derivatives, these satisfy the condition F 0. A solution may dier from a functional solution in the sense that y may not be written explicitly as a function of x. (3) More pictorially, a solution to a dierential equation is a curve in the planeR 2 with the property that the dierential equation holds at all points on the curve. What this means is that at any point on the curve, if we calculate the higher derivatives based on their geometric interpretations, we obtain a bunch of stu that satises the dierential equation. What does this mean and how does this dier from an ordinary equation? Two important dierences: (1) Each solution to an ordinary equation (such as a polynomial equation) is a number. In contrast, each solution to a dierential equation is a function or relation between two variables. (2) When we are looking at an ordinary equation, such asx 2 +x + 2 = 0, we are looking at points where this equation holds. When we are looking at a dierential equation, we are looking at curves such that the equation holds at all points on the curve. (3) To check that an ordinary equation holds at a point, we evaluate at the point. However, to check whether a dierential equation holds, we need to understand behavior locally, on a neighborhood. In other words, it makes no sense to ask whether a dierential equation holds for a given point (x 0 ;y 0 ); it only makes sense to ask whether it holds on a given curve. Basically, a dierential equation seeks to nd a function that exhibits certain local behavior as described by an expression involving the function and its derivatives. Aside: Dierential equations as functional equations. A functional equation is an equation that asks for a function satisfying certain conditions. Specically, a functional equation is an equation in terms of a function that we require to be true for every choice of value for all the letters in the equation, i.e., we require it to be an identity in all letter variables. 1 For instance, the equation: f(x) =f(x)8 x2R has solution set precisely the set of all even functions. Similarly, the equation: f(ax) =af(x)8a;x2R has solution set precisely the set of all functions f of the form f(x) =x, where is a constant. The desired solutions to functional equations are functions, and is does not make sense to ask whether a particular input-output pair satises a functional equation. Dierential equations are a particular kind of functional equations. Specically, dierential equations are functional equations involving derivative behavior all considered at a single point. 2 1 In mathematical jargon, the letter variables for numbers are typically quantied over all integers. 2 There are more complicated functional equations involving derivatives that are not dierential equations in the sense that we have talked about. Examples include delay dierential equations. 2 1.3. Some examples of dierential equations and solutions. We begin with a simple dierential equation: dy dx = 1 We claim that y = x + 13 is a solution to this dierential equation. In the various jargon that we have introduced: (1) The function f(x) =x + 13 satises the condition that f 0 (x) = 1. (2) The curve y =x + 13 satises the condition y 0 = 1. Both these statements are clearly true. Graphically, the curve y =x + 13 is a straight line with slope 1 and intercept 13. Since its slope is 1, dy=dx = 1 everywhere on the line. However, this is not the only solution. Astute observers would have noted that there was nothing partic- ularly auspicious about the number 13. In fact, for any constantC,y =x+C, or the functionf(x) =x+C, solves this dierential equation. Thus, any line with slope 1 is a solution curve to this dierential equation. Note that every value ofC gives one solution, called a particular solution. Each such solution corresponds to a line with slope 1. Pictorially, we get a bunch of parallel lines that cover the entire plane. Are these the only ones? Indeed, which brings us to the next topic. 1.4. De ja vu. When we rst learn algebra in middle or high school, we are given examples such as: How many more apples need to be added to 3 apples to obtain 5 apples? Algebra version: Solve 3 +t = 5. At rst, these seem like silly examples, because anybody who has grasped the concept of subtraction (the inverse operation to addition) can probably solve the problem without any knowledge of algebra. It is only after seeing harder examples of equations that people begin to appreciate the power of algebraic manipulation in solving problems that are too hard to manipulate with basic arithmetic. In the same way, our rst example of a dierential equation is in fact an integration problem. Specically, solving: dy dx = 1 is equivalent to performing the indenite integration: y = Z 1dx which gives the answer y = x +C where C is an arbitrary constant. Each value of C gives a particular solution. Let us elaborate the process a little further: dy dx = 1 Moving the dx to the numerator on the other side, we obtain: dy =dx Note that this is just formal manipulation, akin to the way you learned formal algebra when you got started. Don't ponder about the intrinsic meaning of dy and dx. Integrating both sides, we get: Z dy = Z dx Which gives: y =x +C 3 (Note that we can actually get positive constants with both indenite integrals, but we can absorb the two constants into one). This drawn-out process seems pointless for this specic example, just like algebra seemed pointless when you were solving 3 +t = 5. Unlike middle school, however, where we waded through a lot of these silly examples before getting to more substantive examples of the use of algebra, we can now jump straight to harder situations where we see the machinery of dierential equations and how it gets used. Aside: Yesterday's problem is today's solution. In the elementary grades, addition, subtraction, multiplication and division were the problems, and numbers were the answers. In the middle grades, simple algebraic equations were the problems, and reducing them to a clearly stated arithmetic computation was the crux of the solution (the rest was trivial). When we learned dierentiation, dierentiating functions was the problem, but when we studied graphing and integration, dierentiation was one of the many tools that was used in coming up with solutions. This is the fundamental nature of mathematics: yesterday's focus problems become the taken-for-granted tools of solution to today's focus problems. Reducing the solution to today's focus problem to solving a bunch of yesterday's focus problems is almost as good at solving yesterday's problem. 3 Until very recently in this course, integration was the problem. Now, integration is part of the solution. Once a dierential equation is reduced to calculating a bunch of integrals, we're home. Our goal in nding solution functions to dierential equations is to reduce dierential equations to (one or more) integration problems. 2. Understanding and solving separable equations 2.1. Separable equations: a crash course. A dierential equation of the form: dy dx =f(x) has as solution: y = Z f(x)dx where the right side can be computed by nding one antiderivative and then tacking on a +C. This is a form of dierential equation that directly reduces to a single integration problem { the calculus analogue to 3 +t = 5. Let's look at a slightly more complicated example: dy dx =f(x)g(y) also written as: y 0 =f(x)g(y) In words, we say thaty 0 is a multiplicatively separable function ofx andy { it is the product of a function that depends only on x and a function that depends only on y. We do some algebra-like manipulations whose aim is to bring together on one side all terms involving y and bring together on the other side all terms involving x: dy g(y) =f(x)dx We now put integral signs and carry out indenite integration: Z dy g(y) = Z f(x)dx 3 Assuming you remember how to solve yesterday's problems. Mathematical knowledge is cumulative, but an individual's mathematical knowledge is cumulative only if that individual actually accumulates knowledge. 4 Once we nd antiderivatives, we put the +C on just one side (because two additive constants can be absorbed into one). For instance, consider: dy dx = (x 2 + 1)(y 2 + 4) We proceed to get: Z dy y 2 + 4 = Z (x 2 + 1)dx This becomes: 1 2 arctan y 2 = x 3 3 +x +C What the +C means is that every particular value of C gives a particular solution. For instance, when C = 0, we get the solution: 1 2 arctan y 2 = x 3 3 +x The curve in the plane given by this solution (that we don't need to imagine) satises this dierential equation. Note that although the expression is not in the form of y as a function of x, we can bring it in that form with some algebraic manipulation, to get: y = 2 tan 2x 3 3 + 2x However, this kind of separation and writing things as functions is not always possible (I am glossing over many details here). 2.2. Separable equations from the other side. Suppose we start with a family of curves of the form: F (x) +G(y) =C whereC varies overR. We want to nd a dierential equation that is satised by all curves in the family. We use implicit dierentiation to get: F 0 (x) +G 0 (y)y 0 = 0 which can also be written as: y 0 = F 0 (x) G 0 (y) Which is a (slightly dierently written) version of the original thing we stated out with. Basically, an expression where the derivative y 0 is multiplicatively separable in x and y solves to get a situation where an additively separable function of x and y takes constant values, where each possible constant value gives a particular solution. 2.3. Of circles. Consider, for instance, the family of circles centered at the origin (a concentric family): x 2 +y 2 =a 2 Dierentiating and rearranging terms, we obtain the dierential equation: ydy =xdx In fractions, this becomes: dy dx = x y 5Read More

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