Question

Find the general solution of the differential equation \(\frac{dy}{dx}=\frac{3 \,sec\,⁡y}{2 \,cosec⁡\,x}\).

(a) 3 cos⁡x-2 cos⁡y=C

(b) 3 sin⁡x+2 sin⁡y=C

(c) 3 cos⁡x+2 tan⁡x=C

(d) 3 cos⁡x+2 sin⁡y=C

The question was asked in my homework.

This intriguing question comes from Methods of Solving First Order & First Degree Differential Equations in section Differential Equations of Mathematics – Class 12

Answer 1

Correct choice is (d) 3 cos⁡x+2 sin⁡y=C

The best I can explain: Given that, \(\frac{dy}{dx}=\frac{3 \,sec⁡\,y}{2cosec \,x}\)

\(\frac{2 \,dy}{sec⁡ \,y}=\frac{3dx}{cosec\,⁡x}\)

Separating the variables, we get

2 cos⁡y dy=3 sin⁡x dx

Integrating both sides, we get

∫ 2 cos⁡y dy = ∫ 3 sin⁡x dx

2 sin⁡y=3(-cos⁡x)+C

3 cos⁡x+2 sin⁡y=C