Solve the differential equation: \(x\dfrac{dy}{dx} = x\cdot\tan\left(\dfrac yx\right) + y\).
Q.25 Solve the differential equation: \(x\dfrac{dy}{dx} = x\cdot\tan\left(\dfrac yx\right) + y\).
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Answer: \(\sin\left(\dfrac yx\right) = cx\)
Dividing by \(x\): \(\dfrac{dy}{dx} = \tan\dfrac yx + \dfrac yx\), a homogeneous equation. Put \(y = vx\), \(\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}\).
\(v + x\dfrac{dv}{dx} = \tan v + v \Rightarrow x\dfrac{dv}{dx} = \tan v \Rightarrow \cot v\,dv = \dfrac{dx}{x}\).
Integrating: \(\log|\sin v| = \log|x| + \log|c|\Rightarrow \sin v = cx\).
Hence the general solution is \(\sin\left(\dfrac yx\right) = cx\).
Doubtly AI
· 28 Oct 2026
AI-verified · Oct 2026
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