Solve the given differential equation by separation of variables.
csc(y)dx+sec^(2)(x)dy=0

1 Answers

Best Answer

Step 1
Given:
Given differential equation is
csc(y)dx+sec^(2)(x)dy=0
We want to solve this differential equation by saperation of the variables
Step 2
Calculation:
Given differential equation is
csc(y)dx+sec^(2)(x)dy=0
-> sec^(2)(x)dy=-csc(y)dx
-> (dy)/(csc(y))=-(dx)/(sec^(2)(x))
This is variable saperable form so integrating both side
int (dy)/(csc(y))= -int (dx)/(sec^(2)(x))
Nowcsc (y)=(1)/(sin y),sec(x)=(1)/(cos x)
-> int (sin)dy=-int cos^(2)(x)dx
-> int sin(y)dy=-int (1+cos(2x))/(2)dx
-> -cos(y)=-(1)/(2)(x+(sin(2x))/(2))+c
-> cos(y)=(1)/(2)(x+(sin(2x))/(2))+c' ......(where c'=-c)
This is required solution
Step 3
Answer:
The solution of the differential equati on csc(y)dx+sec^(2)(x)dy=0
is equal to cos(y)=(1)/(2)(x+(sin(2x))/(2))+c'