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Find the equation of a curve passing through (1, π/4) if the slope of the tangent to the curve at any point P (x, y) is y/x – cos2(y/x).
Welcome to AskiitiansAccording 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/xBy 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 = 1Substituting c = 1 in (ii),tan (y/x) + log x = 1Thanks
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