Methods to solve this equation?

Question

The equation is

d2y/dx2 + dy/dx + y = f(x);
y(1) = 3 ; y (3) = 5;
f(x) = 4x^2e^(2x)

Which methods can i use to solve it except
of finding y = x^r e^(2x)(Ax^2+Bx+C)?

Answer 1

First of all -- and leaving initial conditions aside -- if you find ONE solution, then you can find the general solution by adding (to the particular solution) the general solution of the associated homogeneous equation

y'' + y' + y = 0, which you surely know how to solve.

As for the specific one -- yes, guess a form and see what happens.

It's pretty obvious that the base solution to this one would be y(x)=(x^2+Bx+C)e^(2x). So work that through and solve for B and C. Then use the initial conditions to pin down the exact solution you need to the overall equation.