The best source of knowledge for ODEs: video lectures from MIT.
I recommend this book: Differential Equations Problem Solvers
First order, linear, homogeneous equations
1.1 solution Solve the system of ODE's:
1.2 solution Solve
1.3 solution Solve
Second order, linear, homogeneous equations
Second order, linear, nonhomogeneous equations
4.3 solution Find a particular solution of
4.4 solution A ball is thrown straight up from the ground. How high will it go?
The best introduction to the Laplace transform is from an undergrad MIT ODE class, starting at lecture 19 at the link above.
Here are some rules and transforms: http://en.wikipedia.org/wiki/Laplace_transform
6.2 solution Find the Laplace transform of
6.3 solution Find the Laplace transform of