The question is:
We are supposed to find the first 4 terms in the power series. I was able to work through the question but the my answer is wrong, the correct answer is supposed to be
the strange thing is that the answer I got was basically this but the coefficients are square roots of what the answer is, so for example where $1/4$ is, I have $1/2$, and where $1/64$ is, I have $1/8$, etc. I have no idea why this would be the case. I think it could be related to the $r^2=0$ in the roots but even then I have no idea why that would cause all the coefficients to be squared. I'm definitely doing somethign wrong but I don't know what the errors is, the recurrence relation I got was $a_n=-\frac{a_{(n-2)}}{n+r}$
Help will be very much appreciated I've watched many videos on this method and still seem not understand it very well
I suppose that you had a small mistake. Considering $$y=\sum_{n=0}^\infty a_n x^n$$ $$y'=\sum_{n=0}^\infty n a_n x^{n-1}$$ $$y''=\sum_{n=0}^\infty n(n-1) a_n x^{n-2}$$ the differential equation write $$x^2\sum_{n=0}^\infty n(n-1) a_n x^{n-2}+x \sum_{n=0}^\infty n a_n x^{n-1}+x^2\sum_{n=0}^\infty a_n x^n=0$$ that is to say $$\sum_{n=0}^\infty n(n-1) a_n x^{n}+ \sum_{n=0}^\infty n a_n x^{n}+\sum_{n=0}^\infty a_n x^{n+2}=\sum_{n=0}^\infty n^2 a_n x^{n}+\sum_{n=0}^\infty a_n x^{n+2}=0$$ By identification for a given power $m$ of $x$, we then have $$m^2 a_m+a_{m-2}=0$$ with $a_1=0$, $a_2=-\frac 14 a_0$ and so on.