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Flow Defined by a Differential Equation

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Please solve the follwing IVP problem:

Sketch the region Omega in the (t, xo) plane for the initial value problem x'= x^2, x(0) = xo.

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Solution Summary

This solution involves solving an initial value problem and sketching the solution.

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x is a function of t so replace x' with dx/dt. Now, divide both sides by x^2 and multiply both sides by dt.
Integrate to get x = -1/(t + C).

Now, solve for C using,
x(0) = x0 => x0 = -1/C .

Thus, there is a family of solutions.
Note: You should simplify until there is only one fraction i.e. get a common ...

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