Linear ODEs with constant coefficients

We wish to solve a differential equation like $\ddot x + a \dot x + b x = c$. We start by finding the characteristic polynomial, in this case $\lambda^2 + a\lambda + b = 0$.

Our ansatz is $x = A e^{\lambda t}$. We solve the characteristic polynomial for $\lambda$ above and plug it into this solution. There are several cases depending on the roots of $\lambda$.

• If we get distinct real roots for $\lambda$, our life is easy, we just plug them into our solution.

• If we get complex roots, our life is not much harder. When $\lambda = a +ib$, a real solution can be written as $A\, \mathrm{Re}(e^{\lambda t}) + B\, \mathrm{Im}(e^{\lambda t}) = e^a( A \cos bt + B \sin bt)$.

• If a root has multiplicity greater than one, we must write the solution as $(A+Bt)e^{\lambda t}$. This is to ensure we still get the correct number of free parameters based on our initial conditions. Plug in and check that this is a solution, if you are unsure.

These are all solution to the corresponding homogenous ODE, i.e. where $c=0$. Because we have an inhomogenous term, we also have a particular inhomogenous solution. We see that $x=\frac cb$ is such a solution. The general solution is then a superposition of the homogenous and inhomogenous solutions.

One application of 2nd order linear ODEs is finding the behavior damped oscillators.