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According to the given condition,
dy/dx = y/x – cos2(y/x) ………….(i)
This is a homogeneous differential equation.
Substituting y = vx in (i),
v + (x) dv/dx = v – cos2v
⇒ (x)dv/dx = – cos2v
⇒ sec2v dv = – dx/x
By integrating on both the sides,
⇒ ∫sec2v dv = – ∫dx/x
⇒ tan v = – log x + c
⇒ tan (y/x) + log x = c ……….(ii)
Substituting x = 1 and y = π/4,
⇒ tan (π/4) + log 1 = c
⇒ 1 + 0 = c
⇒ c = 1
Substituting c = 1 in (ii),
tan (y/x) + log x = 1
Thanks
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