a.Find the particular solution for the following first-order differential equation with an initial condition by using the method of Separation of Variables.

y'=2ty^{2} y(0)=1

b. Write down a theorem that tells us that the solution to this initial value problem exists and is unique. Verify that the above equation satisfies

the conditions that guarantees there exists a unique solution.

c.

Based on your solution (from part (a)), what is the largest open interval containing the initial condition (the largest interval for t that contains t= 0) for which the solution is well defined (is not undefined, i.e., does not contain a division by zero or a square root of a negative number etc.). Is this consistent with the conclusion of your theorem?

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