Lets say we want to relate the input and the output of a system using a Differential Equation:
since y(t) = x(t) * h(t) , then Y(jw) = X(jw) H(jw) and H(jw) = Y(jw) / X(jw).
Assuming that you have H(jw), lets say H(jw) = A / B, then set that equal to Y(jw) / X(jw).
Now we have A / B = Y(jw) / X(jw).
By cross multiplication we get A X(jw) = B Y(jw).
A and B are expressions made up of $ (jw)^n $ terms.
These correspond to $ d^ny(t)/dt^n $ and $ d^nx(t)/dt^n $terms in the time domain.
Just replace the $ (jw)^n $ terms by the time domain terms from above for all n's to the corresponding functions.
You now have a differential equation that relates the input and the output of a system.
