Find a polynomial f (x) that satisfies the equation xf''(x)+f(x)=x^2

Question

anonymous · Answer

I'd start by looking for the minimal degree polynomial. You can check for yourself that a polynomial of degree less than 2 won't work.
$$f(x)=a_0+a_1x+a_2x^2\
f'(x)=a_1+2a_2x\
f''(x)=2a_2$$
Subbing this into the equation gives
$$2a_2x+a_0+a_1x+a_2x^2=x^2$$
You have to satisfy a few conditions here:
$$\begin{cases}
a_2=1&\text{match up coefficients of quadratic terms}\
a_1+2a_2=0&\text{match up linear coefficients}\
a_0=0&\text{constant coefficients}
\end{cases}$$
This gives $a_0=0$, $a_1=-2$, and $a_2=1$, so one possible polynomial solution is
$$f(x)=x^2-2x$$