Solution of a Differential Equation

 

Definition 4:

The functional relation-ship between the independent variable and the dependent variable (such as y = f(x)) which satisfies the given differential equaion is called the solution of the differential equation.

Example:

Consider the function

f(x) = x² + Ax + B

Clearly (x) is a solution of the differential equation.

y''= 2, because if the second derivative of f(x) gives 2. That is

y = x² + Ax + B is a general solution of the differential equation.

y'' = 2

Each value of A and each value of B (A, B R) gives a particular solution of the differential equation y'' = 2.

Definition 5:

Particular solution of a differential equation

A solution obtained, by assigning particular values to the arbitrary constants in the general solution of the differential equation, is called its particular solution.

Definition 6:

General solution of a differential equation

If the solution of a differential equation of order n contains n arbitrary constants, then it is called the General solution of the differential equation.

Initial Value Problem

Suppose the function

Then

\ f(x) = x² + 2x + 1 is a particular solution of the differential equation y''= 2.

For the function

Because of these condition, the 2^nd order differential equation y''= 2 has particular solution x² + x + 2.

The values f(0) = 2 and f'(0) = 1 are called initial values.

The problem of finding the solution of a differential equation that satisfies these prescribed initial conditions is known as an Initial value problem.

Example:

Show that the function = e^-x+ 2 is the solution of the differential equation y'+ y = 2 , y(0) = 3.

Solution:

If we replace y by , we have

y'+ y

\ f(x) satisfies the equation y' + y = 2

Moreover,