Question

Solve the given initial value problem and determine how the interval in which the solution exists depends on the initial value y0. y' = 5ty², y(0) = y0 y = ?

Answer 1

To solve the given initial value problem y' = 5ty² with y(0) = y0, we can separate variables and integrate both sides. Substituting the initial condition, we can solve for y.

Step-by-step explanation:

To solve the given initial value problem, we have the differential equation y' = 5ty² with the initial condition y(0) = y0.

To solve this, we can separate variables by moving all the y terms to one side and t terms to the other side.

y'/(y²) = 5t => (1/y²)dy = 5tdt

Integrating both sides, we get ∫(1/y²)dy = ∫5tdt.

This simplifies to -1/y = 5t²/2 + C , where C is the constant of integration.

Applying the initial condition y(0) = y0, we get -1/y0 = C.

Substituting this value of C back into the equation, we can solve for y.