Homogeneous Linear ODE Basics (Part 2)

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Last time we covered the steps necessary to get the general solution of a homogeneous linear ODE. This time we'll consider the case where some of the roots of the characteristic equation are complex.

Complex Roots

In a polynomial like the characteristic equation of a homogeneous linear ODE, where the coefficients are real valued, any complex roots will come in conjugate pairs. So if a + bi is a root then a - bi is also a root. Therefore, for one complex conjugate pair of roots, the portion of the general solution will be



using Euler's formula for the first step (which we covered here) and for the second step using the fact that cos(-x) = cos(x) and sin(-x) = -sin(x)

If we want to keep our solutions limited to real valued functions, we can use this formula to generate two linearly independent real-valued solutions. If we let c₀ = c₁ = 1/2, then it becomes



Then if we let c₀ = -i/2 and c₁ = i/2, then we get



So using these two solutions, we can say the part of the general solution cooresponding to this complex conjugate pair of roots is

Repeated Complex Roots

There's one last case we should look at in a bit more depth that most people gloss over. What happens when you have repeated complex roots. We've seen that if you have a repeated complex root with a multiplicity of m (which implies that you have its complex conjugate as a root with multiplicity m as well), then you get this part of the general solution



If we choose specific constants, similar to what we did before in the case of distinct complex roots, we can get a set of real-valued solutions. If for some p in [0, m-1] we let c_{0, p} = c__{1, p} = 1 / 2 and let all other constants be zero, we get



Then letting c_{0, p} = -i/2 and c_{1, p} = i/2, we get



So we can see that with a complex root with multiplicity m, the part of our general solution that corresponds to that root (and the matching complex conjugate root with multiplicity m) is

Conclusion

So now accounting for both real and complex roots, repeated or not, we find that if you factor your characteristic equation, for each factor you have a new term to add in to your general solution.

For each real, distinct root r, add this term



For each real, repeated root r with a multiplicity of m, add this term



For each distinct complex conjugate pair of roots a + bi and a - bi, add this term



And for each repeated complex conjugate pairs a + bi and a - bi, each of which is repeated m times, add this term



or more compactly



Keep in mind that all of the c constants are completely unrelated. If two terms have c₀, for instance, those two constants should be given unique identifiers to keep from confusing the two.

Example

Let's look at one more example just to see this in practice. Let's say we have the homogeneous linear eighth order ODE



which gives us the characteristic equation



And so our general solution is