Mathematics > Differential Equations

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  • An equation involving derivatives of the dependent variable with respect to independent variable (variables) is known as a differential equation.
  • Order of a differential equation is the order of the highest order derivative occurring in the differential equation.
  • Degree of a differential equation is defined if it is a polynomial equation in its derivatives.
  • Degree (when defined) of a differential equation is the highest power (positive integer only) of the highest order derivative in it.
  • A function which satisfies the given differential equation is called its solution.
  • The solution which contains as many arbitrary constants as the order of the differential equation is called a general solution and the solution free from arbitrary constants is called particular solution.
  • To form a differential equation from a given function we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants.
  • Variable separable method is used to solve such an equation in which variables can be separated completely i.e. terms containing y should remain with dy and terms containing x should remain with dx.
  • A differential equation which can be expressed in the form dy/dx = f(x,y) where f(x,y) is a homogenous function of degree zero is called a homogeneous differential equation.
  • A differential equation of the form (dy/dx)+Py = Q, where P and Q are constants or functions of x only is called a first order linear differential equation.

Examples

Question

Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constants.

Solution

We have y = a sin (x + b)

dy/dx = a cos (x + b)…………………………..(1)

d2y/dx2 = – a sin (x + b)……………………….(2)

Eliminating a and b using equations 1 and 2 we get (d2y/dx2) + y = 0.

   

Question

Find the general solution of the differential equation (dy/dx) = (1 + y2)/ (1 + x2)

Solution

Integrating both sides,

   

Question

Find the general solution of the differential equation y dx – (x + 2y2) dy = 0.

Solution

 (dx/dy) – (x/y) = 2y

This is of the type (dx/dy) + Px = Q where P = -(1/y) Q = 2y

Hence IF =

   

The solution of differential equation is

  

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