Consider the initial value problem 2ty′=6y, y(−2)=−8.

a. Find the value of the constant C and the exponent r so that y=C(t^r) is the solution of this initial value problem.
y= ?

b. Determine the largest interval of the form a
c. What is the actual interval of existence for the solution (from part a)?

a.

b.

c.

Step-by-step explanation:

a. Solving as a separable equation.

Divide both sides by y:

Divide both sides by 2t and multiply both sides by dt:

Integrate both sides:

Evaluate the integrals:

Where C is an arbitrary constant.

Solving for y:

Evaluating the initial condition:

So:

b.

Let:

The domain of F(t,y) is:

The domain of    is:

So, Existence and Uniqueness theorem tells us that for each there exists a  unique solution defined in an open interval around

c. The domain of y(t) is:

Hence, the actual interval of existence for the solution y(t) is:

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