Question

How to solve homogeneous second order differential equations

Answer 1

Use Laplace transforms..quickest way to solve them according to me!!!

Answer 2

you have to asign the equation a characteristic polynomial: for example:
2 d2y/dx2 + 4 dy/dx + 6 = 0
2 m2 + 4m +6 =0
and then you solve the equation for m, you'll have two solutions since it is a second order polynomial:
-if m1 = m2 the solution, will have the form Ae^m1*x + Bx e^m1*x
-if m1 is different to m2 and both solutions are real, the solution will have the form
A e^m1*x + B e^m2*x
-if m1 is different to m2 and they both are complex numbers, you use the same form as if they were real numbers, but then you'll have to apply the euler's formula