**Overview**

The method of variation of parameters is the general method which we use to find out a particular solution of a differential equation by replacing the constants in the solution of the homogeneous differential equation by functions and evaluating these functions so that the original DE will be satisfied.

**Method of variation of parameters**

Consider a second order LDE with constant coefficients given by

Then let the complementary function is given by

Then the particular integral is

Where u and v are unknown and to be calculated using the formula

## Solved examples

**Example-1: Solve the following DE by using variation of parameters-**

Sol. We can write the given equation in symbolic form as-

**To find CF-**

It’s AE is

So that CF is –

**To find PI-**

Here

Now

Thus PI will be

So that the complete solution is-

**Example-2: Solve the following by using the method of variation of parameters.**

Sol. This can be written as-

**C.F.-**

Auxiliary equation is-

So that the C.F. will be-

**P.I.-**

Here-

Now

Thus

So that the complete solution is-

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