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Interval of existence of an IVP

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Please solve the following problem using the step by step method:

Find the unique solution (and proof that it is unique) and the maximal interval of the initial value problem x' = x^3, x(0) = xo for any xo E R.

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ODE Problem

We will rewrite the equation as follows:

By integrating on both sides, we get:

Now set t to 0 and get:


So the solution is
There are two theorems which tells us the fact that the solution exists and is unique.
Theorem 1 (Existence)
Suppose that F(x,y) is a continuous function defined in some region

Solution Summary

The expert examines the interval of existence of an IVP.