linear first order differential equation with function and its complex conjugate

Question

this is my first time, I am using math stack exchange, I hope you excuse me for unconventional writing.

I am trying to solve the following differential equation: $$ F\Delta\eta_k(t)+G\Delta\eta_k^*(t)+I=J\partial_t\Delta\eta_k(t)+K\partial_t\Delta\eta_k^*(t) $$. $F,G,I,J,K$ are all real and constant. There are two ideas, I came up with: 1. decomposing the sum and write two differential equations $$ (a) F\Delta\eta_k(t)+\frac{I}{2}=J\partial_t\Delta\eta_k(t)\qquad\qquad (b)G\Delta\eta_k^*(t)+\frac{I}{2}=K\partial_t\Delta\eta_k^*(t) $$ such that $(a)+(b)$ gives me the original equation back. However, this did not give any good result, as they are coupled. 2. The second idea I have, is to write the above equation in terms of only one function $$ \Delta\eta_k(t)=\frac{J}{F}\partial_t\Delta\eta_k(t)+\frac{K\partial_t\Delta\eta_k^*(t)-G\Delta\eta_k^*(t)-I}{F} $$ in order to get the following form $$ f=a\partial_tf+g(t) $$ and then solve for $\Delta\eta_k$. And in using the expression for $\Delta\eta_k$ and take the complex conjugate...Is this the correct way or do you have any better suggestions?

Answer 1

You actually have a pair of independent equations. One on the real part $\rho$ and the other on the imaginary part $\iota$.

$$(F+G)\rho+I=(J+K)\rho'$$ and $$(F-G)\iota=(J-K)\iota'$$

Both equations are elementary.

Hence,

$$\Delta\eta_k(t)=C_\rho e^{(F+G)/(J+K)t}-\frac I{F+G}+i\,C_\iota e^{(F-G)/(J-K)t}.$$

Answer 2

Thank you very much for your quick and very helpful reply Yves Daoust. From your solution, I understand that $\rho\equiv\Re{\Delta\eta_k(t)}=C_\rho e^{(F+G)/(J+K)t}-\frac I{F+G}$ and $\iota\equiv\Im{\Delta\eta_k(t)}=C_\iota e^{(F-G)/(J-K)t}$ However, after plugging in your solution into the RHS of the original differential equation, I get $$ J\partial_t\Delta\eta_k(t)+K\partial_t\Delta\eta_k^*(t)=(F+G)\Delta\eta_k(t)+(\frac{KF}{J}+\frac{JG}{K})\Delta\eta_k(t)^* $$ Hence the $I-$term and the prefactors of $\Delta\eta_k(t)$ and $\Delta\eta_k(t)^*$ are not recovered.

Also, is there a way, where I can have complex exponents?

Edit: I should have said that the real constant coefficients $F,G,I,J,K$ are given