In an ODE class, the differential equation is given
y' + ky = kq_e(t)
where the input $q_e(t)$ is given as $cos \ \omega t$. The teacher "complexifies" the problem by using the real part of $e^{i\omega t}$ due to Euler's formula.
Then we have
y' + ky = k e^{i \omega t}
But since the solution is also complefied, teacher changes notation from $y$, to $\tilde{y}$, hence we now have
\tilde{y}' + k\tilde{y} = k e^{i \omega t}
where the complex solution is $\tilde{y} = y_1 + iy_2$. The claim is we find $\tilde{y}$, then $y_1$ solves the original ODE.
Then teacher goes ahead and solves the problem using exponentials, etc. However what I am looking for is the proof for the statement above, that solving complexified ODE will solve the original ODE.
I tried this
Plug in $\tilde{y} = y_1 + iy_2$ in complexified ODE
(y_1 + iy_2)' + k(y_1 + iy_2) = k e^{i\omega t}
y_1' + iy_2' + ky_1 + kiy_2 = ke^{i\omega t}
Group real #'s and complex #'s together
(y_1+ky_1) + i(y_2' + ky_2) = ke^{i\omega t}
Then the real part of LHS above is $(y_1+ky_1)$, exactly the LHS of primary ODE, and the real part of RHS above is $kcos \ \omega t$ which matches the RHS of original ODE. Is this okay, in terms of language, reasoning, etc.