Question

- Use method of undetermined coefficients...

Answer 1

To solve this:

Answer 2

Here is the solution ->

Answer 3

First find the solution to the homogenous equation (= zero rather than what is on the right hand side)

Answer 4

For the nonhomogeneous solution guess, try \(y_p=Ax^2+Bx+C\):
\[y_p'=2Ax+B\\
y_p''=2A\]
Substitute into the equation:
\[y''+3y'+5y=10x^2-1\\
2A+6Ax+3B+5Ax^2+5Bx+5C=10x^2-1\\
5Ax^2+(6A+5B)x+(2A+3B+5C)=10x^2-1\]
Match up coefficients, solve for \(A,B,C\):
\[\begin{cases}5A=10\\6A+5B=0\\2A+3B+5C=-1\end{cases}\]
Whatever you get for the nonhomogeneous solution will not match up to the solution you're given, however. Check to make sure you posted the right equation/solution.