[Math] Method of Undetermined Coefficients with complex functions as ansatz

Question

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The textbook only mentions the case when ${r}$ is real. What happens when the expression on the right-hand side is complex. For example, I may need to solve the following differential equation:

$${\frac {d^{2}y}{dt^{2}}}+2{\frac {dy}{dt}}+4y= \sqrt{3} e^{-t+\sqrt{3}it}$$

Were I to use the complex ansatz $Ate^{-t+\sqrt{3}it}$, I could successfully solve for A and get the right answer.

1. Why does using complex ansatz work as well?

2. When should I multiply an extra $t$ when using complex anzats? This question concerns $r$ being the possible roots to the auxiliary equation.

Please provide mathematical proof if you can.

Answer 1

1. It is partly answered in this post. The reason why it works, is that real and complex exponential have similar properties with respect to differentiation. Otherwise, just split complex numbers into real part and imaginary part, as suggested in the book and the above-mentioned post.
2. Let us consider the following ansatz: $$ y_p (t) = t^s \sum_{k=0}^m A_k t^k \text{e}^{rt} \, , $$ where $r$ and $A_k$ are complex numbers, $t$ is real, $m$ and $s\in \lbrace 0,1,2\rbrace$ are nonnegative integers.
   • If $s=0$, the computation of the inhomogeneity $a y''_p + b y'_p + c y_p $ gives the condition \begin{aligned} C t^m &= \left(a r^2 + b r + c\right)\sum_{k=0}^m A_k t^k \\ &+ \left(2a r + b\right)\sum_{k=1}^m k A_k t^{k-1} \\ &+ a\sum_{k=2}^m k\left(k-1\right) A_k t^{k-2} \, . \end{aligned} In the right-hand term, the power $t^m$ can be reached if $a r^2 + b r + c \neq 0$, i.e. $r$ is not a root of the auxiliary equation.
   • If $s=1$, one must have \begin{aligned} C t^m &= \left(\left(a r^2 + b r + c\right) t + 2ar+b \right)\sum_{k=0}^m A_k t^k \\ &+ \left(\left(2a r + b\right)t + 2a\right)\sum_{k=1}^m k A_k t^{k-1} \\ &+ at\sum_{k=2}^m k\left(k-1\right) A_k t^{k-2} \, , \end{aligned} which is not possible to solve if $a r^2 + b r + c = 0$ and $2ar + b = 0$. Thus, the same rule as for real $r$ is recovered. The only difference is that the undetermined coefficients $A_k$ may be complex.