How do I work out this problem? The initial condition is $$2u''(t)+2e^{-0.1t}\cdot u'(t)+4e^{-0.2t}\cdot u(t)=0, u(0)=1, u'(0)=0$$
This is what I got after doing some substitution for $u(t),u'(t), u''(t)$ and $e^{\text^}$ using Maclaurin Series. I am supposed to find the first 4 nonzero solutions in the series at about $t=0$
This one of the comments our professor gave us but sure how to apply it to get the solution from my equation.
You can find higher derivatives at $t=0$ by differentiating the differential equation.
Differentiate your differential equation one more time and plug in the known values at $t=0$ to find the $f'''(0)$
Differentiate again and plug in the known derivatives to find $f^{(4)}(0)$ and so forth.
Once you have your derivatives at $t=0$ you can find your series.