Q.34 Solve the differential equation x²dy/dx = x² + xy + y² .

Answer: tan⁻¹(y/x) = log|x| + c dy/dx = 1 + y/x + y²/x² , a homogeneous D.E. Put y = vx , so dy/dx = v + xdv/dx . v + xdv/dx = 1 + v + v² ⇒ xdv/dx = 1 + v² dv/1 + v² = dx/x Integrating: tan⁻¹v = log|x| + c . Replacing v = y/x : tan⁻¹(y/x) = log|x| + c is the general solution.

Solve the differential equation \(x^2\dfrac{dy}{dx} = x^2 + xy + y^2\).

Q.34 Solve the differential equation \(x^2\dfrac{dy}{dx} = x^2 + xy + y^2\).

Doubtly Asked 28 Oct 2026 Board exam · March 2026 hsc 12th Save WhatsApp Report

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Answer: \(\tan^{-1}\left(\dfrac{y}{x}\right) = \log|x| + c\)

\(\dfrac{dy}{dx} = 1 + \dfrac{y}{x} + \dfrac{y^2}{x^2}\), a homogeneous D.E.

Put \(y = vx\), so \(\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}\).

\(v + x\dfrac{dv}{dx} = 1 + v + v^2 \Rightarrow x\dfrac{dv}{dx} = 1 + v^2\)

\(\dfrac{dv}{1 + v^2} = \dfrac{dx}{x}\)

Integrating: \(\tan^{-1}v = \log|x| + c\).

Replacing \(v = \dfrac{y}{x}\): \(\tan^{-1}\left(\dfrac{y}{x}\right) = \log|x| + c\) is the general solution.

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report

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Solve the differential equation \(x^2\dfrac{dy}{dx} = x^2 + xy + y^2\).

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