# Thread: Integral of inhomogenous dif equation

1. ## Integral of inhomogenous dif equation

Hi all

Can someone please help me find the solution to this question please?

2. Originally Posted by moolimanj
Hi all

Can someone please help me find the solution to this question please?
There's a standard process to follow. Where in this process are you stuck?

3. Its the x^2 term thats causing me grief.

Is the solution still of the form: y=p1x+p0

4. Originally Posted by moolimanj
Its the x^2 term thats causing me grief.

Is the solution still of the form: y=p1x+p0
You try to get a particular solution of the form $y = p_2 x^2 + p_1 x + p_0$ where the values of $p_2 \,$, $p_1\,$ and $p_0 \,$ have to be found.

Substitute the particular solution into the DE:

$5 (2 p_2) + 4(2 p_2 x + p_1) + (p_2 x^2 + p_1 x + p_0) = x^2$

$\Rightarrow p_2 x^2 + x(8p_2 + p_1) + (10p_2 + 4p_1 + p_0) = x^2$

Equate coefficients of powers of x:

$p_2 = 1$ .... (1)

$8p_2 + p_1 = 0$ .... (2)

$10p_2 + 4p_1 + p_0 = 0$ .... (3)

So a particular solution is $y = x^2 - 8x + 22$.

Now add this particular solution to the homogenous solution to get the general solution.