What is the meaning of having imaginary solutions to a differential equation I am trying to solve this equation
$$x+e^{xt}=0$$
for different values of x and see when there is no real solution. Plotting this on a graph and assuming x is real gives me the different values of t. I want to find the value of t when the above equation has no real solution. To do this, I first differentiate the equation and set that value equal to $0$. This gives me the relation for the minimum value of $$x+e^{-xt}$$. When the value of t is greater than this the curve doesnt cut the x axis and there is no real solution. But now assuming x is imaginary I can still solve to get a solution which consists of complex terms. But I dont understand what this means in the real domain. If there is no real value of $x$ for which there is a solution, then what does allowing imaginary values of $x$ do? How does a system like this work in real life?

In cases where the result represent an impedance, the real part is the ohmic resistance, independent of frequency. The imaginary part is either capacitive or inductive depending of the sign.
Complex numbers are used for systems that can store then release energy after some time.
