In this theorem we identify that for a specified differential equation a set of fundamental solutions will exist.

Consider the differential equation

 y′′ + p (t ) y′ + q (t ) y = 0

 Here p(t) and q(t) are continuous functions on any interval I. select t₀ to be any point in the interval I. Let y₁(t) be a solution to the differential equation which satisfies the initial conditions.

 y(t₀) = 1

 y′ (t₀) = 0

 Let y₂(t) be a solution to the differential equation which satisfies the initial conditions.

 y (t₀) = 0

 y′ (t₀) = 1

 So y₁(t) and y₂(t) form a fundamental set of solutions for the differential equation.

This is easy enough to illustrate that these two solutions form a fundamental set of solutions. Just calculate the Wronskian.

Thus, fundamental sets of solutions will exist; we can solve the two IVP's specified in the theorem.