Question

Solve the differential equation:

cosec³ x dy − cosec y dx = 0

Answer 1

Given, differential equation is

cosec³ x dy − cosec y dx = 0

⇒ cosec³ x dy = cosec y dx

⇒ `int "dy"/("cosec y") = int "dx"/("cosec"^3 "x")`

⇒ `int sin "y dy" = int sin^3 "x dx"`

⇒ `- cos "y" = int sin^2 "x". sin "x dx"`

= `int (1 - cos^2 "x"). sin "x dx"`

Let cos x = t,

⇒ − sin x dx = dt

∴ − cos y = `- int (1 - "t"^2) "dt"`

⇒ cos y = `"t" - "t"^3/3 + "c"`

∴ cos y = `cos "x" - (cos^3 "x")/3 + "c"`