Suppose we want to find a power series solution to this common physics-based ODE Initial Value Problem: Just need the first 5 or so terms, not the full general solution. $$-x'' = -kx' -sinx$$ $$x(0)=0 \; \text{and} \; x'(0)=1$$
My Process:
Ansatz: $$\sum_{n=0}^{\infty}a_nt^n$$ Firstly, observe that based on our initial values, this tells us that $a_0=0$ and $a_1=1$.
Plugging this into our ODE, we have: $$\implies -\frac{d^2}{dt^2} \left[ \sum_{n=0}^{\infty}a_nt^n \right] = -k\frac{d}{dt} \left[ \sum_{n=0}^{\infty}a_nt^n \right] -\sin \left( \sum_{n=0}^{\infty}a_nt^n \right)$$
$$\implies -\left[\sum_{n=2}^{\infty}a_{n-2}(n)(n-1)t^{n-2} \right] = -k \left[ \sum_{n=1}^{\infty}a_{n-1}(n)t^{n-1} \right] -sin \left( \sum_{n=0}^{\infty} a_nt^n \right) $$
$$\implies 0= \left[\sum_{n=2}^{\infty}a_{n-2}(n)(n-1)t^{n-2} \right] -k \left[ \sum_{n=1}^{\infty}a_{n-1}(n)t^{n-1} \right] - \left[ \sum_{m=0}^{\infty} \frac{(-1)^m}{(2m+1)} \left( \sum_{n=0}^{\infty} a_n t^n \right)^{2m+1} \; \right] $$
This is clearly too messy to work with, so I had the idea of making my ansatz just power 7 let's say if I wanted my solution to be up to power 5. However, I assume the expansion of this would also be computationally too intensive. What am I missing here? Is there an easier Ansatz incorporating matrices to make this computationally doable???
