Particular Solution to Non-homogeneous Second Order DE

[BOAS]

Homework Statement

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.

Homework Equations

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.

 

[ehild]

Homework Statement

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.

Homework Equations

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.

Your solution is correct, c=10.5.

 

[Mark44]

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

What's going on here?

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.