The first thing you should do is take an introductory course in Differential Equations!
I also should point out that I don't like "Laplace Transform" methods for solving differential equations. There are, in my opinion, always simpler ways.
"Laplace Transform" methods do have the "advantage" that they are very "mechanical"- there is relatively little thinking required!
The Laplace transform of a function, f(x), is defined as \(\displaystyle L(f)(s)= \int_0^\infty e^{-st}f(t)dt\).
By using "integration by parts", you can show that the Laplace Transform of the derivative of f, L(f'(x)), is s times the Laplace Transform of f minus f(0), L(f'(x))= sL(f)- f(0), that the Laplace Transform of the second derivative of f. L(f''(x)) is s^2 times the Laplace Transform of f minus s times f(0 ) minus f'(0), L(f''(x))= s^2L(f)- sf(0)- f'(0).
So, for example, if we have the initial value problem, ay''+ by'+ cy= g(x), y(0)= a, y'(0)= b, Taking the Laplace transform of both sides gives the algebraic equation as^2L(y)- asf(0)- af'(0)+ bsL(y)- bf(0)+ L(y)= (as^2- bs+ c)L(y)- asf(0)- af(0)- bf(0)= L(g). Solve that algebraically for L(y) and look up y in a table of "inverse Laplace transforms".
