Consider the wave equation $$ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} $$ for $(t,x) \in \mathbb{R^2}$
We know that $u(0,x) = \varphi_0(x)$ and $\frac{\partial u}{\partial t} = \varphi_1(x)$ for $x \in \mathbb{R}$ in which $\varphi_0,\varphi_1$ are real analytic functions, which means they are locally given by a convergent power series.
i) determine a solution to the Cauchy Problem in form of a power series $$ u(t,x)= \sum_{k=0}^{\infty} \frac{\partial^k }{\partial t^k} u(0,x) \frac{t^k}{k!} $$ I have done this part of the problem. I have the power series: $$ u(t,x) = \sum_{k=0}^{\infty} \frac{\partial^k }{\partial t^k} u(0,x) \frac{t^k}{k!} = \sum_{k=0}^{\infty} \varphi_0^{(2k)}(x)\frac{t^{2k}}{(2k)!} + \sum_{k=0}^{\infty} \varphi_1^{(2k)}(x) \frac{t^{2k+1}}{(2k+1)!} $$ I need help with:
- How to prove that this power series is a solution of the given wave equation?
- How to know, if the power series is convergent ? because a priori it is unclear whether the series is convergent
Thanks in advance