First I found the first to derivatives of the following power series:




I used these equations to rewrite my initial equation:


After making all the summations the same ($n=0$), I got the following:


I finally got to the following step, but I'm stuck here not knowing how I can factor out the $x$ so I can find the first non-zero terms:



To solve the differential equation, you must write it in the form $$\sum_{n=0}^\infty b_nx^n=0$$ so you must put together the same powers of $x$.

For instance, you may want to use more convenient indexing, such as \begin{align}\left(\sum_{n=0}^\infty b_nx^n\right)'&=\sum_{n=0}^\infty (n+1)b_{n+1}x^n\\\left(\sum_{n=0}^\infty b_nx^n\right)''&=\sum_{n=0}^\infty (n+2)(n+1)b_{n+2}x^n\\ x^{\color{red}{k}}\cdot\sum_{n=0}^\infty b_nx^n&=\sum_{\color{red}{n=k}}^\infty b_{n\color{red}{-k}}x^n\end{align}

Using these should put you in the right direction.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.