Question

Solve the initial value problem and determine where the solution attains its maximum value. y' =(2-e")/(3+2y), y(0)=0

Answer 1

To solve the initial value problem y' =(2-e")/(3+2y), y(0)=0, we can separate variables and integrate both sides. The solution is y = ln(3+2t). The maximum value of y occurs at t = 1 and y = ln(5).

Step-by-step explanation:

To solve the initial value problem y' =(2-e")/(3+2y), y(0)=0, we can separate variables and integrate both sides.

1. Start by multiplying both sides of the equation by (3+2y) to get rid of the denominator.
2. Next, rearrange the terms so that all the y-related terms are on one side and the t-related terms are on the other side.
3. Now, we can integrate both sides with respect to t, remembering to add the constant of integration.
4. Solve for y in terms of t to obtain the general solution.
5. Finally, use the initial condition y(0)=0 to find the specific solution.

The solution to the initial value problem y' =(2-e")/(3+2y), y(0)=0 is y = ln(3+2t).

To determine where the solution attains its maximum value, we can take the derivative of y with respect to t and set it equal to zero to find the critical points. In this case, the maximum value occurs when t = 1 and y = ln(5).