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

User Gbeaven
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Final answer:

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.

User Patrick Hurst
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