Ordinary Differential Equations/Applications to Linear Equations

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Existence of Solutions[edit]

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.