Question

Use the method for solving equations of the form dy/dx = G(a+by) to solve the following differential equation: dy/dx = 3 sin(15-5y)

Answer 1

To solve the differential equation dy/dx = 3 sin(15-5y), you can separate the variables, integrate, and solve for y. The solution is y = 3 - (1/5) tan^(-1)(-5(x + K) - 5C + 75) + (1/5) tan^(-1)(-5K + 75).

Step-by-step explanation:

To solve the differential equation dy/dx = 3 sin(15-5y) using the method for solving equations of the form dy/dx = G(a+by), we can follow these steps:

1. Separate the variables by moving all terms involving y to one side and all terms involving x to the other side.
2. Integrate both sides with respect to x.
3. Solve the resulting integral equation for y.

In this case, we have dy/dx = 3 sin(15-5y). Separating the variables gives us (1/3) sec^2(15-5y) dy = dx. Integrating both sides gives us (1/3)(-1/5) tan(15-5y) + C = x + K, where C and K are constants of integration. Solving for y gives us y = 3 - (1/5) tan^(-1)(-5(x + K) - 5C + 75) + (1/5) tan^(-1)(-5K + 75).