"Solve the given differential equation by separation of..." was answered by our experts

Efan Halliday

Answered question

2021-10-21

Solve the given differential equation by separation of variables.

\csc y d x + \sec^{2} x d y = 0

doplovif

Skilled2021-10-22Added 71 answers

\csc y d x + \sec^{2} x d y = 0

Rearrange terms

\sec^{2} x d y = - \csc y d x

Separate the variables

\frac{d y}{\csc y} = - \frac{1}{\sec^{2} x} d x

\sin y = - \cos^{2} x d x

Integrate both sides

\int \sin y = - \int \cos^{2} x d x

Use the identity

\cos^{2} x = \frac{1 + \cos 2 x}{2}

, so

\int \sin y = - \int (\frac{1 + \cos 2 x}{2}) d x

Distribute

\int \sin y = - \int (\frac{1}{2} + \frac{\cos 2 x}{2}) d x

\int \sin y = - \frac{1}{2} \int d x - \frac{1}{2} \int \cos 2 x d x

Rewrite

\int \sin y = - \frac{1}{2} \int d x - \frac{1}{2 (2)} \int \overset{\cos u}{\overset{⏞}{\cos 2 x}} \overset{d u}{\overset{⏞}{(2) d x}}

Apply

\int \sin u d u = - \cos u + C and \int \cos u d u = - \sin u + C

- \cos y = - \frac{1}{2} x - \frac{1}{4} \sin 2 x + C

Multiply both sides by -4

4 \cos y = 2 x + \sin 2 x + C

Result:

4 \cos y = 2 x + \sin 2 x + C

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