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

1. ## 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 $x^m$ satisfy the following D.E.? $3x^2(\frac{d^2y}{dx^2}) + 11x(\frac{dy}{dx})-3y=0$

So my strategy is to just substitute $x^m$ for y and find the values of m such that the equation equals 0.
$y=x^m$
$y'=mx^{m-1}$
$y''=m(m-1)x^{m-2}$

After substituting;

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

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

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

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

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

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

2. ## 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.