Finding the constant for the particular solution to $y''(x) + y(x) = 2^x$

Question

I'm really confused on this problem. Right now, I'm solving for the particular solution of:

$y''(x) + y'(x) = 2^x$

My test solution was $A^x$, and I got

$yp(x) = 2^x/(ln(A)^2 + 1)$

My problem here is that the solution in the book makes it so that A equals 2. I have no idea how they solved for A. Could anyone point me in the right direction?

Answer 1

The "test solution" should be $a \cdot 2^x$, not $A^x$. You want the same exponential multiplied by a constant coefficient.