Find the green function for the following BVP by using dirac delta function and solve the BVP using Green function

Question

Here's the question:

$y'' + y = x^2 + 1, y(0) = 5, y(1) = 0$

I manage to get a solution to be this:

$y = -6\cot(1)\sin(x) + 6\cos(x) + x^2 - 1 - 2\cos(1)\sin(x) + 2\cot(1)\sin(1)\sin(x)$

Can somebody help me check if my $y_p$ is correct?

Thanks!

Answer 1

First of all, the last two terms of your solution cancel out since $\cot(1)\sin(1) = \cos(1)$.

Then, $$y_p = -6\cot(1)\sin(x) + 6\cos(x) + x^2 - 1$$ $$y'_p = -6\cot(1)\cos(x) + 6\sin(x) + 2x$$ $$y''_p = 6\cot(1)\sin(x) - 6\cos(x) + 2$$ Thus, $y''_p + y_p = x^2 +1 $, and the boundary conditions are also satisfied, $$y(0) = -6\cot(1)\sin(0) + 6\cos(0) + x^2 - 1 = 5$$ $$y(1) = -6\cot(1)\sin(1) + 6\cos(1) + 1^2 - 1 = 0$$ So yes, it works.