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Question
Solve the differential equation:
cosec3 x dy − cosec y dx = 0
Solution
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"`
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