My electrical engineering class just moved into differential equations from linear algebra, which is a topic I've never touched on before. The professor doesn't work hardly any examples on the board, so I've found myself a bit confused on how to solve differential equations that aren't linear. Below I've attached a homework problem that I'm having quite a bit of trouble solving. Is there any way I could get some help with the steps on how to solve an exponential differential equation (in simplistic terms too, please!) from an example in this homework problem? I don't want the entire problem answered, just one or two with an explanation so I can wrap my head around how to solve these things. Thanks!
Problem:
Best Answer
For a constant-coefficient linear ODE, in this case one of the form $y’’+ay’+by=0$, solutions can be easily obtained without working backwards from the solutions. The first step is to compute the roots of the characteristic polynomial $r^2+ar+b=0$. Let the roots of this equation be $r_1,r_2$. Then solutions to the differential equation take the form $y_1=e^{r_1x},y_2=e^{r_2x}$. If the root is repeated (i.e. $r_1=r_2$, the solutions are $y_1=e^{r_1x},y_2=xe^{r_1x}$. If the roots are complex, use Euler’s formula $e^{it}=\cos t+i\sin t$ to simplify solutions.