For which values of m does x^m satisfy the D.E.? Just wanting a double check.

Since I kept getting this wrong last night I thought I should double check with ya'll to make sure I'm doing everything correctly. Thanks.

For which values of m, does $\displaystyle x^m$ satisfy the following D.E.? $\displaystyle 3x^2(\frac{d^2y}{dx^2}) + 11x(\frac{dy}{dx})-3y=0$

So my strategy is to just substitute $\displaystyle x^m$ for y and find the values of m such that the equation equals 0.

$\displaystyle y=x^m$

$\displaystyle y'=mx^{m-1}$

$\displaystyle y''=m(m-1)x^{m-2}$

After substituting;

$\displaystyle 3x^2(m(m-1)x^{m-2}) + 11x(mx^{m-1})-3(x^m)=0$

there's a cancellation in the first two terms, since $\displaystyle x^{m-2}=\frac{x^m}{x^2}$

$\displaystyle 3(m(m-1)x^{m} +11mx^m -3x^m=0$

$\displaystyle x^m(3m^2-3m+11m-3)=0$

so the discriminant is $\displaystyle b^2-4ac=8^2-4*3*-3=100$

$\displaystyle (-8+10)/2*3=1/3$ and $\displaystyle (-8-10)/2*3=-3$

Re: For which values of m does x^m satisfy the D.E.? Just wanting a double check.

Yes, m= 1/3 and m= -3 are correct.