# Integral of inhomogenous dif equation

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• Feb 15th 2008, 01:44 AM
moolimanj
Integral of inhomogenous dif equation
Hi all

Can someone please help me find the solution to this question please?
• Feb 15th 2008, 01:49 AM
mr fantastic
Quote:

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?
• Feb 15th 2008, 02:48 AM
moolimanj
Its the x^2 term thats causing me grief.

Is the solution still of the form: y=p1x+p0
• Feb 15th 2008, 03:00 AM
mr fantastic
Quote:

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 \$\displaystyle y = p_2 x^2 + p_1 x + p_0\$ where the values of \$\displaystyle p_2 \,\$, \$\displaystyle p_1\, \$ and \$\displaystyle p_0 \,\$ have to be found.

Substitute the particular solution into the DE:

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

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

Equate coefficients of powers of x:

\$\displaystyle p_2 = 1\$ .... (1)

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

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

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

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