# Homework Help: Particular Solution to Non-homogeneous Second Order DE

1. Dec 19, 2015

### BOAS

1. The problem statement, all variables and given/known data
Find a particular solution to
$y'' - 3y' + 2y = 6x^2$

I don't understand how/why the value of c has been determined. I'm hoping it is a mistake in the solution, but knowing me, it's probably my mistake.

2. Relevant equations

3. The attempt at a solution

assume a solution of the form $y = ax^2 + bx + c$, then

$y' = 2ax + b$ and

$y'' = 2a$

Subbing this into our DE gives

$2a - 6ax - 3b + 2ax^2 + 2bx + 2c = 6x^2$.

$a = 3$ to complete the $x^2$ terms.

$b = 9$ to cancel the $x$ term.

and I think $c = 10.5$ but the solution states that $c = 2.5$

What's going on here?

Thanks.

2. Dec 19, 2015

### ehild

3. Dec 19, 2015

### Staff: Mentor

It's very easy to check, and something that you should always do.
Substitute your solution, y = 3x2 + 9x + 10.5 into the differential equation to see if you get 6x2. Doing this should convince you that your solution is correct. If you do the same with the textbook's solution, you can see that it is incorrect.