Q.23 Solve the D.E. dy/dx + y/x = x³ - 3

Answer: xy = x⁵/5 - 3x²/2 + c This is linear, dy/dx + Py = Q with P = 1/x , Q = x³ - 3 . I.F. = e^(∫ 1/xdx) = e^(log x) = x . Solution: y· x = ∫ (x³ - 3)x dx + c = ∫ (x⁴ - 3x) dx + c therefore xy = x⁵/5 - 3x²/2 + c , i.e. y = x⁴/5 - 3x/2 + c/x .

Solve the D.E. \(\frac{dy}{dx} + \frac{y}{x} = x^3 - 3\)

Q.23 Solve the D.E. \(\frac{dy}{dx} + \frac{y}{x} = x^3 - 3\)

Doubtly Asked 28 Oct 2026 Board exam · July 2025 hsc 12th Save WhatsApp Report

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Answer: \(xy = \frac{x^5}{5} - \frac{3x^2}{2} + c\)

This is linear, \(\frac{dy}{dx} + Py = Q\) with \(P = \frac{1}{x}\), \(Q = x^3 - 3\).

I.F. \(= e^{\int \frac{1}{x}dx} = e^{\log x} = x\).

Solution: \(y\cdot x = \int (x^3 - 3)x\,dx + c = \int (x^4 - 3x)\,dx + c\)

\(\therefore xy = \frac{x^5}{5} - \frac{3x^2}{2} + c\), i.e. \(y = \frac{x^4}{5} - \frac{3x}{2} + \frac{c}{x}\).

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

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Solve the D.E. \(\frac{dy}{dx} + \frac{y}{x} = x^3 - 3\)

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