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

# Lagrange's method for a PDE

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Chaitra (new course), 4 marks
- Video: [Lagrange's Method to solve Partial Differential Equation | Msc Mathematics #mathematics](https://www.youtube.com/watch?v=MYSqmQ03l7A) — Spectrum of Mathematics
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Solve the PDE z(p − q) = z² + (x + y)² using Lagrange's method, where p = ∂z/∂x and q = ∂z/∂y.

## Short answer

Auxillary equations give x+y = C1 and (z²+(x+y)²) e^{-2x} = C2. General solution: φ(x+y, (z²+(x+y)²)e^{-2x})=0, i.e., z²+(x+y)² = e^{2x} f(x+y) with arbitrary f.

## Answer

Lagrange's linear PDE is Pp + Qq = R. Its auxiliary (subsidiary) equations are dx/P = dy/Q = dz/R. Here P = z, Q = −z, R = z² + (x + y)².

**Auxiliary equations**

dx/z = dy/(−z) = dz/[z² + (x + y)²]

**First integral.** From the first two ratios, dx = −dy, so dx + dy = 0 and

x + y = a   ... (1)

**Second integral.** Use dx/z = dz/(z² + a²) with (x + y) = a, so z dz/(z² + a²) = dx.

Integrating: ½ ln(z² + a²) = x + c, that is ln(z² + a²) − 2x = 2c, so

(z² + (x + y)²) e^(−2x) = b   ... (2)

**General solution.** With an arbitrary function φ:

**φ( x + y, (z² + (x + y)²) e^(−2x) ) = 0**

Equivalently, z² + (x + y)² = e^(2x) f(x + y) for an arbitrary function f.

Check: take u = (z² + (x+y)²)e^(−2x). Then z·u_x − z·u_y + (z² + (x+y)²)u_z = 0, as required for a solution of the auxiliary system.

## Check yourself

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

1. For Pp + Qq = R:
   A. dx/p = dy/q
   B. dx·P = dy·Q
   C. dx/P = dy/Q = dz/R
   D. P dx = R dz only
   Answer: C. Standard. (Key point: Auxiliary equations)

2. From dx/z = dy/(−z):
   A. x − y = a
   B. xy = a
   C. z = a
   D. x + y = a
   Answer: D. dx + dy = 0. (Key point: First integral)

3. The second integral is:
   A. (z^2 + (x+y)^2) e^(−2x) = b
   B. (z^2 + (x+y)^2) e^(2x) = b
   C. z + x + y = b
   D. z^2 − (x+y)^2 = b
   Answer: A. Integrating z dz/(z^2 + a^2) = dx. (Key point: Second integral)

4. The general solution is:
   A. φ(x+y, z) = 0
   B. φ(x + y, (z^2 + (x+y)^2)e^(−2x)) = 0
   C. φ(x, y) = 0
   D. z = x + y
   Answer: B. Arbitrary function of the two integrals. (Key point: General solution)

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