# Math Help - Integral of inhomogenous dif equation

1. ## Integral of inhomogenous dif equation

Hi all

2. Originally Posted by moolimanj
Hi all

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.