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 t_{0} to be any point in the interval I. Let y_{1}(t) be a solution to the differential equation which satisfies the initial conditions.
y(t_{0}) = 1
y′ (t_{0}) = 0
Let y_{2}(t) be a solution to the differential equation which satisfies the initial conditions.
y (t_{0}) = 0
y′ (t_{0}) = 1
So y_{1}(t) and y_{2}(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.