I have a differential equation of the form below:
$$k_1\frac{df(t)}{dt} + (k_2 + k_3e^{k_4 f(t)})f(t) = x(t)$$
I want to find the solution of $f(t)$ as a function of $x(t)$. In other words, I want to find the dynamic response of $f(t)$ to $x(t)$.
First, I tried to use Laplace transforms to find something that looks like a transfer function, and then simply convert it back into time domain. However, I notice that the exponent term makes it a bit complicated. So I decided to first simplify the equation using some substitution for the exponential term. I know that $$\frac{de^{f(t)}}{dt} = e^{f(t)} \frac{df}{dt}$$
But I can't seem to be able to apply this information. Can someone please help me here?
For a nonlinear equation, there is no simple "dynamic response": it is unlikely that you can get anything like a closed form for the dependence of the solution on $x(t)$. In this case I don't even think there's a closed form for the solution in the case $x(t)=t$. Laplace transforms are unlikely to be helpful for nonlinear differential equations.