# 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 , 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.