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

# Charpit's method for a PDE

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2081 Chaitra (new course), 4 marks
- Video: [15. Charpit's Method | Complete Concept & Problem#1 | PDE | Most Important Problem](https://www.youtube.com/watch?v=xdXFBF33bOc) — MKS TUTORIALS by Manoj Sir
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/charpits-method-for-a-pde-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Apply Charpit's method to solve the PDE p²x + q²y = z, where the symbols have their usual meanings.

## Short answer

Using Charpit’s equations from F=p²x+q²y−z=0, we get dp/(p−1)=−dx/(2x) → p=1+a/√x. Then substituting and integrating dz = p dx+q dy gives the complete integral z = (√x + a)² + (√y + b)² with arbitrary constants a,b.

## Answer

For F(x, y, z, p, q) = 0, Charpit's auxiliary equations are

dp/(Fₓ + pF_z) = dq/(F_y + qF_z) = dz/(pF_p + qF_q) = dx/(−F_p) = dy/(−F_q)

Here F = p²x + q²y − z. So Fₓ = p², F_y = q², F_z = −1, F_p = 2px, F_q = 2qy.

**Find a relation.** Take dp/(Fₓ + pF_z) = dx/(−F_p):

dp/(p² − p) = −dx/(2px) → dp/(p − 1) = −dx/(2x)

Integrating: ln(p − 1) = −½ ln x + ln a, so p − 1 = a/√x, that is

p = 1 + a/√x

**Find q.** Put p in the PDE: q²y = z − p²x = z − (√x + a)², so

q = √( [z − (√x + a)²] / y )

**Integrate dz = p dx + q dy.** Let w = z − (√x + a)². Then dw = dz − (1 + a/√x) dx = dz − p dx = q dy:

```text
dw = √(w/y) dy       →   dw/√w = dy/√y
2√w = 2√y + 2b       →   √w = √y + b
```

**Complete integral**

**z = (√x + a)² + (√y + b)²**

where a and b are arbitrary constants.

Check: p = (√x + a)/√x and q = (√y + b)/√y, so p²x + q²y = (√x + a)² + (√y + b)² = z.

## Check yourself

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

1. In Charpit's method, the equation dp/(Fx + pFz) is paired with which other ratio to get a relation for p?
   A. dz/(pFp + qFq)
   B. dx/(-Fp)
   C. dy/(-Fp)
   D. dx/(Fq)
   Answer: B. Charpit's system includes dp/(Fx+pFz) = dx/(-Fp), used to find p here. (Key point: Charpit's equations)

2. For F = p^2 x + q^2 y - z, what are Fx, Fp and Fz?
   A. p^2, 2px, 1
   B. p^2, 2px, -1
   C. p^2, 2p, 1
   D. 2px, p^2, -1
   Answer: B. Differentiate F with respect to x, p and z respectively. (Key point: Partial derivatives)

3. Integrating dp/(p-1) = -dx/(2x) gives p equal to?
   A. a/sqrt(x)
   B. 1 + a/sqrt(x)
   C. 1 - a/x
   D. 1 + a sqrt(x)
   Answer: B. ln(p-1) = -1/2 ln x + ln a, so p - 1 = a/sqrt(x). (Key point: Finding p)

4. What is the complete integral?
   A. z = a sqrt x + b sqrt y
   B. z = (sqrt x + a)^2 + (sqrt y + b)^2
   C. z = (sqrt x + a)^2 - (sqrt y + b)^2
   D. z = (x + a)^2 + (y + b)^2
   Answer: B. Integrating dz = p dx + q dy gives the sum of two squares with constants a and b. (Key point: Complete integral)

## Close variants set in other exams

- [Charpit's method for a PDE](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Acharpits-method-for-a-pde) — 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-two) — Mathematics III, chapter 4, asked 3×