I'm looking for a proof of the Laplace Transform method of solving differential equations.

I have several books that discuss the Laplace Transform and how to apply it, but no proof of how it works.

As an example of a simple differential equation that can be solved using Laplace, consider the mass on a spring with damping. We know what the solutions are (equations for growing or decaying oscillations), but how does Laplace make this possible?

 

I'm not sure what you are looking for in terms of a proof. Heuristically you can start with the proofs of the existence an uniqueness of the solutions to differential equations. Once you know that a solution exists and that it is unique, any method you use to arrive at that unique solution can be used.

 

I'm looking for a proof of the Laplace Transform method of solving differential equations.

I have several books that discuss the Laplace Transform and how to apply it, but no proof of how it works.

As an example of a simple differential equation that can be solved using Laplace, consider the mass on a spring with damping. We know what the solutions are (equations for growing or decaying oscillations), but how does Laplace make this possible?

Hello there,

One way would be to use the Fourier Transform and Inverse Fourier Transform as a starting point, deriving the Laplace Transform from that. The starting point would be to transform the function e^(-r*t)*v(t) rather than just v(t) (where r here is 'rho') and eventually you could work it into a multiplication of e^(-s*t)*v(t) inside the integral (s=rho+j*w), which is the Laplace Transform.

 

I'm looking for a proof of the Laplace Transform method of solving differential equations.

I have several books that discuss the Laplace Transform and how to apply it, but no proof of how it works.

As an example of a simple differential equation that can be solved using Laplace, consider the mass on a spring with damping. We know what the solutions are (equations for growing or decaying oscillations), but how does Laplace make this possible?

The Laplace Transform is an operator; it operates on an input and produces an output. Namely, differential time-varying systems to linear algebraic systems. What you can do is use analysis to prove definitions and theorems in order to construct let's say a "mathematical toolbox".

 

Less a rigorous proof than an explanation: