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**First-Order Differential Equations**

**Definition****(i) Order of a differential equation**

Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation. Consider the following differential equations:

dy / dx = e^{x}

The first, second and third equations involve the highest derivative of first, second and third order respectively. Therefore, the order of these equations are 1, 2 and 3 respectively.**(ii) Degree of a differential equation**

To study the**degree of a differential equation**, the key point is that the differential equation must be a polynomial equation in derivatives, i.e., y′, y″, y″′ etc. Consider the following differential equations:

We observe that first equation is a polynomial equation in y″′, y″ and y′ with degree 1, second equation is a polynomial equation in y′ (not a polynomial in y though) with degree of 2. Degree of such differential equations can be defined. But third equation is not a polynomial equation in y′ and degree of such a differential equation can not be defined.

By the degree of a differential equation, when it is a polynomial equation in derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation.**Note:**Order and degree (if defined) of a differential equation are always positive integers.**Example****Find the order and degree, if defined, of each of the following differential equations:**

(i) dy/dx - cosx = 0

(ii) xy d^{2}y/dx^{2}+ x(dy/dx)^{2}- ydy/dx = 0

(ii) y' + y^{2}+ e^{y}= 0**Solution****(i)**The highest order derivative present in the differential equation is dy / dx, so its order is one. It is a polynomial equation in y′ and the highest power raised to is dy / dx one, so its degree is one.**(ii)**The highest order derivative present in the given differential equation is d^{2}y / dx^{2}, so its order is two. It is a polynomial equation in d^{2}y / dx^{2}and dy/dx and the highest power raised to d^{2}y / dx^{2}is one, so its degree is one.**(iii)**The highest order derivative present in the differential equation is y′′′, so its order is three. The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined.**Separable Equations****(i) Separable differential equation***y' = A(x)B(y)***Example:**

RC circuits: Charging:

Discharging:

IR = Q/C ⇒ Q = Q_{o}e^{-1/RC}**(ii)****Linear Differential Equations**

y'(x) + p(x)y(x) = q(x)

integrating factor:**Example:**

y' = 3x^{2}- y/x

y(1) = 5**(iii) Exact Differential Equations****(a)****Potential function:**For**(b)****Exact differential equation:**A potential function exists; general solution: ∅(x, y) =**Example:****Theorem:**Test for exactness: dM / dy = dN / dx**Example:**x^{2}= 3xy + (4xy + 2x)y'= 0

e^{x}sin y - 2x + (e^{x}cosy + 1)y' = 0**(iv) Integrating Factors****(a) Integrating factor:**μ(x, y) ‡ 0 such that μM(x,y) + μN(x,y)y'*= 0*is exact.**Example:**y^{2}- 6xy + (3xy - 6x^{2})y' = 0**(b) How to find integrating factor:****Example:**x - xy - y'= 0**A.**Separable equations and integrating factors: μ= 1/B**B.**Linear equations and integrating factors:**(v) Homogeneous and Bernoulli Equations****(a)**Homogeneous differential equation: ; let*y = ux*⇒ separable.**Example:**.**(b) Bernoulli equation:**y' = P(x)y = R(x)y^{α}; α= 0 ⇒ linear;α= 1

⇒ separable; otherwise, let v = y^{1-}^{α}⇒ linear**Example:**.

**Higher order linear odes**

**Homogeneous linear odes**

An linear ordinary differential equation of order n is said to be homogeneous if it is of the form

a_{n}(x)y^{(n)}+ a_{n - 1}(x)y^{(n - 1)}+ .... + a_{1}(x)y' + a_{0}(x)y = 0

where y'= dy/dx i.e., if all the terms are proportional to a derivative of y (or y itself) and there is no term that contains a function of x alone.

However, there is also another entirely different meaning for a first-order ordinary differential equation. Such an equation is said to be homogeneous if it can be written in the form

dy/dx = F(y/x)**Points to Remember****(i)**F(x,y,y', ....y^{(n)}) = 0, a*n*th order ODE if the*n*th derivative of the unknown function y(x) is the highest occurring derivative.**(ii)****Linear ODE:**y^{(n)}+ p_{n - 1}(x)y^{(n - 1)}+ ...+ p_{1}(x)y' + p_{0}(x)y = g(x)**(iii)****Homogeneous linear ODE:**y^{(n)}+ p_{n - 1}(x)y^{(n - 1)}+ ...+ p_{1}(x)y' + p_{0}(x)y = g(x) = 0**Theorem:**Fundamental Theorem for the Homogeneous Linear ODE: For a homogeneous linear ODE, sums and constant multiples of solutions on some open interval*I*are again solutions on*I*. (This does not hold for a nonhomogeneous or nonlinear ODE!).**General solution:**y= c_{1}y_{1}+...+ c_{n}y_{n}, where y_{1},....,y_{n}is a basis (or fundamental system) of solutions on*I*; that is, these solutions are linearly independent on*I*.**(iv)****Linear independence and dependence:**n functions y_{1},....,y_{n}are called linearly independent on some interval*I*where they are defined if the equation k_{1y1},....,k_{n}y_{n = 0}_{ }on*I*implies that all k_{1},....,k_{n}are zero. These functions are called linearly dependent on*I*if this equation also holds on*I*for some k_{1},....,k_{n}not all zero.

Example: . Sol.: y = c_{1}e^{-2x}+ c_{2}e^{-x}+ c_{3}e^{x}+ c_{4}e^{2x}**Theorem:**Let the homogeneous linear ODE have continuous coefficients p_{0}(x),... , p_{n-1}(x) on an open interval*I*. Then*n*solutions y_{1},....,y_{n}on*I*are linearly dependent on*I*if and only if their Wronskian is zero for some x= x_{0}in*I*. Furthermore, if*W*is zero for x= x_{0}, then*W*is identically zero on*I*. Hence if there is an in*I*at which*W*is not zero, then y_{1},....,y_{n}are linearly independent on*I*, so that they form a basis of solutions of the homogeneous linear ODE on*I*.

Wronskian:**(v)****Initial value problem:**An ODE with n initial conditions y(x_{0}) =K_{0}, y'(x_{0}) = K_{1}, y_{n-1}(x_{0}) = K_{n}.**Homogeneous linear odes with constant coefficients****(i)**: Substituting y= e^{λx}, we obtain the characteristic equation λ^{n}+ a_{n-1}λ^{n-1}+ ...+ a_{1}λ + a_{0}= 0.**(a)****Distinct real roots:**The general solution is**Example:**y''' - 2y'' - y' + 2y =0.**Solution:**y= c_{1}e^{-x}+ c_{2}e^{x}+ c_{3}e^{2x}**(b)****Simple complex roots:**, y_{1}= e^{px }cos(qx), y2 = e^{px sin}(qx) .**Example:**y''' - y'' - 100y' + 100y =0.**Solution:**y = c_{1}e^{x}+ c_{2}cos10x + c3 sin10x.**(c)****Multiple real roots:**If λ is a real root of order*m*, then*m*corresponding linearly independent solutions are: e^{λx},xe^{λx}, x^{2}e^{λx},x^{m-1}e^{λx}.**Example:**y^{(5)}- 3y^{(4)}+ 3y''' - y'' = 0.**Solution:**y = c_{1}+ c_{2}x + (c_{3 }+ c_{4}x + c_{5}x2)e^{x}.**(d****) Multiple complex roots:**If are complex double roots, the corresponding linearly independent solutions are:

e^{px}cos(qx), e^{px}sin(qx),xe^{px}cos(qx), xe^{px}sin(qx), .**(ii)**Convert the higher-order differential equation to a system of first-order equations.**Example:**.**Non - Homogeneous linear odes****(i)**y^{(n)}+ p_{n-1}(x)y^{(n-1)}+ ...+ p_{1}(x)y' + p_{0}(x)y = g(x) the general solution is of the form: y = y_{h}+ y_{p}where y_{h}is the homogeneous solution and y_{p }is a particular solution.**(ii) Method of undermined coefficients****Example:****y'''+ 3y'' + 3y'+y = 30e****Solution:**y = (c_{1}+ c_{2}x + c_{3}x^{2})e^{-x }+ 5x^{3}e^{-x}**(iii) Method of variation of parameters:**

y_{p = u1y1 +...+ unyn}

where , k = 1,...., n.**Example:**.**Solution:**.

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