Question

What is a general solution to the differential equation y′=sinxcscy?

Answer 1

The general solution to the differential equation y' = sin(x)csc(y) is found by separating variables and integrating, resulting in -ln|csc(y) + cot(y)| = -cos(x) + C.

Step-by-step explanation:

The question is asking for the general solution to the differential equation y′=sin(x)csc(y). To find the general solution, we separate the variables and integrate both sides. This involves moving all terms involving y to one side of the equation and all terms involving x to the other side. To solve this specific equation, follow these steps:

1. Separate the variables: csc(y) dy = sin(x) dx.
2. Integrate both sides: The integral of csc(y) with respect to y is -ln|csc(y) + cot(y)|, and the integral of sin(x) with respect to x is -cos(x).
3. Combine the constants of integration into a single constant C.
4. The general solution is then given by -ln|csc(y) + cot(y)| = -cos(x) + C.

Note that this is a simplified process. The actual integration and algebra may be more complex depending on the initial conditions and the domain of the solution.