Question

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

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

Answer 1

To solve the given separable differential equation for y using the substitution u = t + 25y and dy/dt (t + 25y)², you can follow these step-by-step instructions.

Step-by-step explanation:

To solve the separable differential equation for y by making the substitution u = t + 25y and dy/dt (t + 25y)², you can follow these steps:

1. Take the derivative of u with respect to t using the chain rule: du/dt = 1 + 25(dy/dt).
2. Replace dy/dt with (du/dt - 1) / 25 in the original equation.
3. Simplify the equation and separate the variables: (du/dt - 1)(t + 25y)² = 25y.
4. Integrate both sides of the equation with respect to u and t.
5. Solve the resulting equation to find the solution for u in terms of t.
6. Substitute u back into the equation u = t + 25y to solve for y.
7. Apply the initial condition y(0) = 9 to find the value of the constant of integration.