Second order linear ODE

Second order linear ODEs are in the following form:

$y''+p(t)y'+q(t)y=g(t)$

Important types of second order linear ODEs include

• Homogeneous
• Constant coefficients (where p and q are constants)

Initial value problem

There are two arbitrary constants in the solution of a second order linear ODE, so we need two initial conditions.

${\begin{cases}y(t_{0})=y_{0}\\y'(t_{0})=y_{0}'\end{cases}}$

Solutions

Constant coefficient, homogeneous

Whatever derived usually also works for variable coefficients.

These are the simplest kind. They have the general form

$ay''+by'+cy=0$

We can guess the form of the solution to be exponential, since exponential functions has the property of being the same after many differentiation.

$y=e^{rt}$

We substitute in the guess and obtain the characteristic equation

${\begin{aligned}ar^{2}e^{rt}+bre^{rt}+ce^{rt}&=0\\ar^{2}+br+c&=0\end{aligned}}$

Depending on the constants, it will give us anywhere from zero to two solutions: $y_{1}$ and $y_{2}$

There are three cases: 2 real roots, complex conjugates, and repeated roots, depending on the discriminant.

For 2 real roots, the two solutions are always independent so we get general solution easy.

For complex conjugates,

For repeated roots,

Fundamental set of solutions

Given that two linearly independent solutions are given, the general solution is of the following form

$y(t)=c_{1}y_{1}+c_{2}y_{2}$

The independence of the solutions can be checked using the Wronskian. The two solutions are called a fundamental set of solutions.

This fundamental set always exists according to the General existence theorem.