---
title: "Lagrange's method for a PDE"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:lagranges-method-for-a-pde-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/lagranges-method-for-a-pde-three
site: BCT Engineering
---

# Lagrange's method for a PDE

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2081 Chaitra (new course), 4 marks
- Video: [method of multipliers || lagrange's partial differential equations|| problem 12](https://www.youtube.com/watch?v=1t-xO-9fX9A) — HAMEEDA MATHTUBER
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/lagranges-method-for-a-pde-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Solve the partial differential equation (y − z)p + (x − y)q = (z − x) using Lagrange's method.

## Short answer

The auxiliary equations dx/(y−z)=dy/(x−y)=dz/(z−x) give d(x+y+z)=0 so x+y+z = constant. Using multipliers (2x−y−z,2z−x−y,2y−x−z) yields an exact differential, integrating to 2x²−y²−z²−2xy−2xz+4yz = constant. General solution: φ(x+y+z, 2x²−y²−z²−2xy−2xz+4yz)=0.

## Answer

**Reading of the question.** The equation is solved exactly as printed: (y − z)p + (x − y)q = z − x.

The auxiliary equations are dx/(y − z) = dy/(x − y) = dz/(z − x).

**First integral.** Add the numerators and denominators: the sum of the denominators is (y − z) + (x − y) + (z − x) = 0, so dx + dy + dz = 0 and

x + y + z = a

**Second integral.** Use the multipliers l = 2x − y − z, m = 2z − x − y, n = 2y − x − z. Then

```text
l(y − z) + m(x − y) + n(z − x) = 0
```

(expanding, every term cancels). So l dx + m dy + n dz = 0, which is exact:

(2x − y − z) dx + (2z − x − y) dy + (2y − x − z) dz = 0

Integrating (it is the differential of ½[2x² − y² − z² − 2xy − 2xz + 4yz]):

2x² − y² − z² − 2xy − 2xz + 4yz = b

**General solution**

**φ( x + y + z, 2x² − y² − z² − 2xy − 2xz + 4yz ) = 0**

where φ is an arbitrary function. (Verified by direct substitution: both functions satisfy (y − z)uₓ + (x − y)u_y + (z − x)u_z = 0.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What are Lagrange's auxiliary equations for (y-z)p + (x-y)q = z-x?
   A. dx/(y-z) = dy/(x-y) = dz/(x-z)
   B. dx/p = dy/q = dz/(z-x)
   C. dx/(y-z) = dy/(x-y) = dz/(z-x)
   D. dx/(z-x) = dy/(y-z) = dz/(x-y)
   Answer: C. Coefficients of p, q and the right side give the denominators. (Key point: Auxiliary equations)

2. Why does adding the numerators and denominators give x + y + z = a?
   A. The sum of denominators is zero, so dx+dy+dz = 0
   B. dx = dy = dz
   C. The denominators are all equal
   D. Each denominator equals x+y+z
   Answer: A. (y-z)+(x-y)+(z-x) = 0, so dx+dy+dz = 0 and integrating gives x+y+z = a. (Key point: First integral)

3. What must hold for the multipliers l, m, n to give a second integral?
   A. l x + m y + n z = 0
   B. l + m + n = 0
   C. l(y-z) + m(x-y) + n(z-x) = 0
   D. l = m = n
   Answer: C. Then l dx + m dy + n dz = 0 and it can be integrated. (Key point: Multipliers)

4. What form does the general solution take?
   A. u + v = constant
   B. phi(u, v) = 0 with u, v the two integrals and phi arbitrary
   C. u = v
   D. Only u = a
   Answer: B. The general solution links the two independent integrals through an arbitrary function. (Key point: General solution)

## Related in Mathematics III

- [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
- [Lagrange's method for a PDE](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alagranges-method-for-a-pde-two) — Mathematics III, chapter 4, asked 3×
- [Lagrange's method for a PDE](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alagranges-method-for-a-pde) — Mathematics III, chapter 4
- [Classify a second order PDE](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aclassify-a-second-order-pde) — Mathematics III, chapter 4
- [Solve an integral equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-an-integral-equation) — Mathematics III, chapter 4

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