When we get an exponential on the RHS of our differential equation (or if we give an exponential input) it's quite easy to solve the equation, but if we provide a sinsusoidal or cosinusoidal input, the calculation becomes horrendous. So we go for a trick, we give an imaginary, complex exponential input and solve the differential equation as usual by assuming \$A \, e^{i \omega t}\$ is the solution. And finally take the real part for a cosinsusoidal input and imaginary part for sinusoidal input (superposition theorem). The only big thing here is finding the complex amplitude \$A\$, once we find \$A\$ we can just multiply it with \$ e^{i\omega t}\$ take real part and arrive at the solution. Here there's a neat trick, when we observe the response in a simple circuit, say an RC circuit we are measuring the voltage across the capacitor. The voltage across the cap is so very similar to an Voltage divider but with a slight change, instead of \$R_2\$ it has a \$\frac {1}{i\omega C}\$
So whenever we are given a circuit with capacitances and inductances, we convert it into an abstract circuit, considering only the complex amplitudes. Find the required amplitude and multiply \$ e^{i\omega t}\$
Now the Laplace transform somehow performs all these neat little tricks in a neat little integral. But I do not understand how, rather I can't relate these two processes eventhough they are the same.