I recently went across the IVP and decided to explore it for my class. Though I still don't get how to get the solution. I have worked some of it up but I'm stuck, does anyone know how to continue?
Given: $$\dot{x}=Ax$$
With solution: $$x(t)=e^{At}x(0)$$
My workings: \begin{align} \dot{x} &= Ax \\ \dot{x}-Ax& = 0 \\ e^{-At}\dot{x}-Ae^{-At}x& = 0\\ e^{-At}\dot{x}-\frac{d}{dt}(Ae^{-At}x)&=0 \\ Ae^{-At}x&=0 \end{align}
Then I'm stuck, did I do this correctly or did I miss anything?