Key to "Solve the given differential equation by separation of..."

Ronin Tran

Open question

2022-08-20

Solve the given differential equation by separation of variables.

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

historia075c

Beginner2022-08-21Added 6 answers

Step 1
Given:
Given differential equation is

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

We want to solve this differential equation by saperation of the variables
Step 2
Calculation:
Given differential equation is

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

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

\Rightarrow \frac{d y}{\csc (y)} = - \frac{d x}{\sec^{2} (x)}

This is variable saperable form so integrating both side

\int \frac{d y}{\csc (y)} = - \int \frac{d x}{\sec^{2} (x)}

Nowcsc

(y) = \frac{1}{\sin y}, \sec (x) = \frac{1}{\cos x}

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

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

\Rightarrow - \cos (y) = - \frac{1}{2} (x + \frac{\sin (2 x)}{2}) + c

\Rightarrow \cos (y) = \frac{1}{2} (x + \frac{\sin (2 x)}{2}) + c^{'} \dots \dots {w h e r e c^{'} = - c}

This is required solution
Step 3
Answer:
The solution of the differential equati on

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

is equal to

\cos (y) = \frac{1}{2} (x + \frac{\sin (2 x)}{2}) + c^{'}

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