# How can I solve for the general solution of a second ODE using Laplace transform?

I am interested in solving for the most general solution, in other words with the constants c1, c2, etc. Take the first example from Paul's Notes:

http://tutorial.math.lamar.edu/Classes/DE/IVPWithLaplace.aspx

Let's say it did not give me the initial conditions and it just wanted a general solution. How could I solve for this general solution?

Thanks!

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Since solving the linear ODEs by using Laplace transform should involve the terms of $y^{[n]}(0)$ , if appears, don't panic, just let them to some arbitrary symbols.
However in those situations, Laplace transforms are often not the best choice, since inverse Laplace transforms are often very difficult to solve, so there is a type of the evolving method for homogenous linear ODEs, no need to involve inverse transforms, just involve the choice of constants instead, is so called the "kernel method". For example the homogenous linear ODEs of the type $\sum\limits_{k=0}^n(a_kx+b_k)y^{(k)}(x)=0$ , quite a lot of the cases can be solved by assuming the integral kernel of the form $y=\int_Ce^{xs}K(s)~ds$ . Particular solution to a Riccati equation $y' = 1 + 2y + xy^2$ and Help on solving an apparently simple differential equation are the good examples.