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Revision Notes on Differential Equations

  • The order of the differential equation is the order of thederivative of the highest order occurring in the differential equation.
  • The degree of a differential equation is the degree of the highest order differential coefficient appearing in it subject to the condition that it can be expressed as a polynomial equation in derivatives.
  • A solution in which the number of constants is equal to the order of the equation is called the general solution of a differential equation. 
  • Particular solutions are derived from the general solution by assigning different values to the constants of general solution.
  • An ordinary differential equation (ODE) of order n is an equation of the formF(x, y, y',….., y(n) ) = 0, where y is a function of x and y' denotes the first derivative of y with respect to x.
  • An ODE of order n is said to be linear if it is of the form an(x)y(n) + an-1(x) y(n-1) + …. + a1(x) y' + a0 (x) y = Q(x)
  • If both m1 and m2 are constants, the expressions (D–m1) (D–m2) y and (D–m2) (D–m1) y are equivalent i.e. the expression is independent of the order of operational factors.
  • A differential equation of the form dy/ dx = f (ax+by+c) is solved by writing ax + by + c = t.
  • A differential equation, M dx + N dy = 0, is homogeneous if replacement of x and y by λx and λy results in the original function multiplied by some power of λ, where the power of λ is called the degree of the original function.
  • Homogeneous differential equations are solved by putting y = vx.
  • Linear equation are of the form of dy/dx + Py = Q, where P and Q are functions of x alone, or constants.
  • Linear equations are solved by substituting y =uv, where u and v are functions of x.
  • The general method for finding the particular integral of any function is 1/ (D-α)x = eαx∫Xe-αxdx

Various methods of finding the particular integrals:
1. When X = eax in f(D) y = X, where a is a constant
Then 1/f(D) eax = 1/f(a) eax , if f(a) ≠ 0 and
1/f(D) eax = xr/fr(a) eax , if f(a) = 0, where f(D) = (D-a)rf(D)
2. To find P.I. when X = cos ax or sin ax
f (D) y = X
If f (– a2) ≠ 0   then 1/f(D2) sin ax = 1/f(-a2) sin ax
If f (– a2) = 0 then (D2 + a2) is at least one factor of f (D2)
3. To find the P.I.when X = xm  where m ∈ N
f (D) y = xm
y = 1/ f(D) xm
4. To find the value of 1/f(D) eax V where ‘a’ is a constant and V is a function of x
1/f (D) .eax V = eax.1/f (D+a). V
5. To find 1/f (D). xV where  V is a function of x
1/f (D).xV = [x- 1/f(D). f'(D)] 1/f(D) V

Some Results on Tangents and Normals:
1. The equation of the tangent at P(x, y) to the curve y= f(x) is Y – y =  dy/dx .(X-x)
2. The equation  of the  normal  at point P(x, y)  to the  curve y = f(x) is Y – y =  [-1/ (dy/dx) ].(X – x )
3. The length of the tangent  = CP  =y √[1+(dx/dy)2]
4. The  length of the normal = PD = y √[1+(dy/dx)2]
5. The length of the Cartesian sub tangent  = CA = y dy/dx
6. The length of the Cartesian subnormal = AD = y dy/dx
7. The initial ordinate of the tangent = OB = y – x.dy/dx

The document Revision Notes: Differential Equations | Calculus - Mathematics is a part of the Mathematics Course Calculus.
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FAQs on Revision Notes: Differential Equations - Calculus - Mathematics

1. What is a differential equation?
Ans. A differential equation is an equation that relates a function with its derivatives. It represents a mathematical model that describes how a function changes over time or in relation to other variables.
2. What are the types of differential equations?
Ans. There are several types of differential equations, including ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve functions of a single variable, while PDEs involve functions of multiple variables.
3. How do you solve a differential equation?
Ans. The method of solving a differential equation depends on its type and order. There are various techniques such as separation of variables, integrating factors, and using series solutions that can be employed to solve differential equations.
4. What are the applications of differential equations?
Ans. Differential equations have numerous applications in various fields of science and engineering. They are used to model physical phenomena such as population growth, fluid flow, heat transfer, electrical circuits, and quantum mechanics, to name a few.
5. Can differential equations be solved numerically?
Ans. Yes, differential equations can be solved numerically using methods like Euler's method, Runge-Kutta methods, and finite difference methods. These numerical techniques approximate the solution by discretizing the differential equation and solving it iteratively.
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