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Homogeneous Equations with Constant Coefficients: Complex Roots
Suppose that the equation
$ ay''+by'+c=0 $
which has solutions of the form
$ y=e^{rt} $
has complex roots in its characteristic equation (see the previous section)
$ r_1=\lambda+i\mu; \qquad r_2=\lambda-i\mu $
According to the formula given in the previous section, solutions to the differential equation will be of the form
$ \begin{align} y&=c_1e^{r_1t}+c_2e^{r_2t}\\ &=c_1e^{(\lambda+\mu i)t}+c_2e^{(\lambda-\mu i)t} \end{align} $
To some, this expression might look somewhat incoherent, given the presence of complex numbers in the exponentials. Although it's not strictly necessary, and although there's nothing terribly wrong with this answer, it can be transformed into something a little more familiar. It turns out, the preceding expression is actually equivalent to
$ y=c_1e^{\lambda t}cos\,\mu t+c_2e^{\lambda t}sin\,\mu t $
Now, you genuinely should be familiar with the derivation of this formula. It requires the use of Euler's formula, which you may remember from calculus, and the fact that the solution space of the differential equation is a linear space. But in practice, you'll usually just be using the above formula. As soon as you solve the characteristic equation of the differential equation and find the complex roots, this formula can be used.
(This page under construction)
