Question

Find the solution of the following differential equation:

(?^3 y/dx^3)-7(d^2 y/dx^2)+10(dy/dx)=e^2x sinx

Answer 1

Given,

----------------(1)

First, find the solution to the homogeneous equation.

The auxiliary equation is  

Hence the solution of the complementary equation is,

--------------------(2)

where c₁, c₂, and c₃ are constants.

Now we have to find the particular solution.

Using the method of undetermined coefficients, assume the particular solution as

------------(3)

Now y_p(x) satisfies equation (1),

Find

Again differentiating we get,

Again differentiating the above equation we get,

Substituting the values of in (1) we get,

Equating the coefficients of like terms of x in both sides we get,

Solving the above equations we get,

Hence from (3),

Hence the solution of the given differential equation (1) is