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