Ordinary Differential Equations/Applications to Linear Equations

Existence of Solutions

Just like with separable equations, not all initial value problems for linear equations have a solution.

Theorem 1: If P(x) and Q(x) are continuous on an interval I containing the point $x_{0}$, then the initial value problem has a single unique solution.

This is different from separable equations where the conditions for uniqueness and existence are different - with linear equations, if it exists, it will be unique.

Proof We will use the method of successive approximations just as we did before.