--- title: "Solve p³−4xyp+8y²=0" subject: "Mathematics I" chapter: 4 key: "semester_1/mathematics_i:de-p3-4xyp-8y2" source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-p3-4xyp-8y2 site: BCT Engineering --- # Solve p³−4xyp+8y²=0 Mathematics I · Chapter 4: Ordinary Differential Equations and their Applications - Asked in: 2068 Shrawan - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-p3-4xyp-8y2 - Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4 ## Question Find the general solution of the differential equation: p³ − 4xyp + 8y² = 0. ## Short answer Solve p³ − 4xyp + 8y² = 0 for x: x = p²/(4y) + 2y/p. General solution: 64y = c(4x−c)². Singular solution (envelope): y = 4x³/27. ## Answer p³ − 4xyp + 8y² = 0, with p = dy/dx. It is solvable for x. #### Solve for x 4xyp = p³ + 8y² → x = p²/(4y) + 2y/p #### Differentiate w.r.t. y dx/dy = 1/p: 1/p = p/(2y) · dp/dy − p²/(4y²) + 2/p − (2y/p²) dp/dy Collect the terms: 0 = (dp/dy)[p/(2y) − 2y/p²] − p²/(4y²) + 1/p p/(2y) − 2y/p² = (p³ − 4y²)/(2yp²), and 1/p − p²/(4y²) = −(p³ − 4y²)/(4y²p). Therefore (p³ − 4y²)[ (1/(2yp²)) dp/dy − 1/(4y²p) ] = 0 #### General solution The second factor: dp/dy = p/(2y), so dp/p = dy/(2y), giving **p² = cy**. Substitute p³ = p · cy into the given equation: p(cy − 4xy) + 8y² = 0 → p(c − 4x) = −8y → p = 8y/(4x − c) Put this in p² = cy: 64y²/(4x − c)² = cy, so **64y = c(4x − c)²** (Check: p = c(4x − c)/8 and p² = c²(4x − c)²/64 = cy ✓.) #### Singular solution The first factor p³ = 4y² with the equation gives 12y² = 4xyp, so p = 3y/x. Then 27y³/x³ = 4y², so **y = 4x³/27** It also satisfies the differential equation, and it is the envelope of the family 64y = c(4x − c)². ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. p^3 - 4xyp + 8y^2 = 0 is solved for: A. None; use factorisation B. x, then differentiate w.r.t. y C. y, then differentiate w.r.t. x D. p, then integrate Answer: B. x = p^2/(4y) + 2y/p. (Key point: Approach) 2. The differentiated equation factors as: A. (p^3 + 4y^2)[...] = 0 B. (p - 2y)[...] = 0 C. (p^3 - 4y^2)[...] = 0 D. p(p^2 - 4y)[...] = 0 Answer: C. Both groups of terms contain p^3 - 4y^2. (Key point: Factor structure) 3. The general solution is: A. 64y = c(4x - c)^2 B. 64y = c(4x + c)^2 C. y = c(4x - c) D. 16y = c(x - c)^2 Answer: A. From p^2 = cy and p = 8y/(4x - c). (Key point: General solution) 4. The singular solution is: A. y^2 = 4x^3/27 B. y = 4x^3/27 C. y = x^3 D. y = 27x^3/4 Answer: B. p^3 = 4y^2 combined with the equation gives p = 3y/x and y = 4x^3/27. (Key point: Singular solution) ## Close variants set in other exams - [Solve xy(p²+2)=2py³+x³](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-xy-p2-plus-2) — Mathematics I, chapter 4, asked 2× - [Solve (D²+2D+1)y=eˣ cos x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-2d-1-ex-cosx) — Mathematics I, chapter 4, asked 2× ## Related in Mathematics I - [Solve (1+y²)+(x−e^(tan⁻¹y))dy/dx=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-y2-x-minus-e-atan-y) — Mathematics I, chapter 4, asked 6× - [Solve dy/dx = y/x + tan(y/x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-dy-dx-y-over-x-tan) — Mathematics I, chapter 4, asked 6× - [Solve dy/dx − 2y tan x = y² tan x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-bernoulli-2y-tan-x) — Mathematics I, chapter 4, asked 6× - [Solve (1+x²)dy/dx+y=e^(tan⁻¹x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-x2-dy-dx-etan) — Mathematics I, chapter 4, asked 6× - [Solve x dy/dx + y log y = xy eˣ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-x-dy-dx-y-log-y) — Mathematics I, chapter 4, asked 4× - [Solve x log x dy/dx + y = 2 log x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-x-log-x-dy-dx) — Mathematics I, chapter 4, asked 4× ## Related in other subjects - [Solve a partial differential equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-a-partial-differential-equation) — Mathematics III, chapter 4 - [Solve a partial differential equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-a-partial-differential-equation-two) — Mathematics III, chapter 4 - [LC oscillation differential equation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Alc-oscillation-differential-equation) — Engineering Physics, chapter 1 - [Series solution of Legendre's equation](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aseries-solution-legendre-s-equation) — Mathematics, chapter 4, asked 4×