Second Order Linear

Second Order Linear Ordinary Differential Equations follow the general form

Let , , and be continuous and on the open interval then there exists a unique solution for the Initial Value Problem, and

Also called the superposition principle for homogenous equations
If and are two solutions of a homogenous equation

Then is also a solution

If and are two solutions of a homogenous equation and

Then is the general solution

Let and be two solutions of a homogenous equation, the Wronskian of the equation can be given as

If then and are linearly dependent
If then and are linearly independent

Theorem II vs Theorem III

While both theorems give a similar form, theorem III gives the general solution while theorem II only gives a single solution

Let and be two solutions of a homogenous equation

If the Wronskian of the solutions at is not zero

For the initial condition: and
There exists a unique choice of and such that

Proof for the condition that