Question

Solve the separable differential equation for y by making the substitution u = t + 25y.

dt/dy= (t + 25y)².
Use the following initial condition: y(0) = 9.
Note: Use arctan(x) for the inverse tangent function.

Answer 1

The solution to the separable differential equation is obtained by substitution, separating variables, integrating both sides, applying initial conditions, and back substitution to solve for y in terms of t.

Step-by-step explanation:

The question requires solving a separable differential equation for y with the substitution u = t + 25y. Given that dy/dt = (t + 25y)² with initial conditions y(0) = 9, a step-by-step method involves:

1. Making the substitution to transform the equation into one involving u and t.
2. Separating variables to isolate terms involving u on one side and terms involving t on the other.
3. Integrating both sides to find a solution in terms of u.
4. Using the initial condition to solve for the integration constant.
5. Substituting back to obtain the solution in terms of y and t.

Recall that y(t) represents the position of an object at time t and that the slope of a curve at a particular point is equal to the slope of the tangent line at that point. In the context of the problem, y may represent displacement or position in a physics context.