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

# Charpit's method for a PDE

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Chaitra (new course), 4 marks; 2082 Kartik (new course), 4 marks (asked 2 times)
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Apply Charpit's method to solve the PDE: 2zx − px² − 2qxy + pq = 0, where the symbols have their usual meanings.

## Short answer

Using Charpit's method, from dq/0 we get q = a (constant). Then p = 2x(ay - z)/(a - x^2). Integrating dz = p dx + q dy yields the complete integral z = ay + b(a - x^2), where a and b are arbitrary constants.

## Answer

**Charpit's method.** For F(x, y, z, p, q) = 0 (p = ∂z/∂x, q = ∂z/∂y), find a second relation f(x, y, z, p, q) = a from the auxiliary equations

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

then solve for p and q and integrate dz = p dx + q dy.

**Given:** F = 2zx − px² − 2qxy + pq = 0.

#### Step 1: Partial derivatives
```text
F_p = −x² + q          F_q = −2xy + p
F_x = 2z − 2px − 2qy   F_y = −2qx      F_z = 2x
```

```text
F_x + pF_z = 2z − 2px − 2qy + 2px = 2z − 2qy
F_y + qF_z = −2qx + 2qx = 0
```

#### Step 2: A simple relation
From dq/(F_y + qF_z) = dq/0, we get dq = 0, so

```text
q = a  (constant)
```

#### Step 3: Find p
Put q = a in F = 0:

```text
2zx − px² − 2axy + pa = 0
p (a − x²) = 2x (ay − z)
p = 2x (ay − z)/(a − x²)
```

#### Step 4: Integrate dz = p dx + q dy
```text
dz = 2x (ay − z)/(a − x²) dx + a dy
d(z − ay) = − 2x (z − ay)/(a − x²) dx
```

Let u = z − ay. Then du/u = −2x dx/(a − x²) = d[ln(a − x²)], so ln u = ln(a − x²) + ln b:

```text
z − ay = b (a − x²)
```

**Complete integral: z = ay + b(a − x²)**, with arbitrary constants a and b.

**Check:** p = −2bx, q = a. Then 2zx − px² − 2qxy + pq = 2x(ay + ab − bx²) + 2bx³ − 2axy − 2abx = 0 ✓.

## Check yourself

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

1. Charpit's equations include dx/F_p = dy/F_q = dz/(pF_p+qF_q) =
   A. dp/(F_x) = dq/(F_y)
   B. -dp/F_p = -dq/F_q
   C. dp/(F_y+pF_z) = dq/(F_x+qF_z)
   D. -dp/(F_x+pF_z) = -dq/(F_y+qF_z)
   Answer: D. Standard. (Key point: Auxiliary equations)

2. Here F_y + qF_z =
   A. 0
   B. 2x
   C. -2qx
   D. 2z - 2qy
   Answer: A. -2qx + q(2x) = 0, so dq = 0. (Key point: Simplification)

3. Therefore
   A. q = x
   B. q = a, a constant
   C. q = p
   D. q = 0 only
   Answer: B. dq=0. (Key point: Value of q)

4. z =
   A. ax + b(a - y^2)
   B. ay + b(a + x^2)
   C. ay + b(a - x^2)
   D. ay - b(a - x^2)
   Answer: C. From z - ay = b(a - x^2). (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-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-two) — Mathematics III, chapter 4, asked 3×

## Related in other subjects

- [Solve y = px − √(m²+p²)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-clairaut-y-px-sqrt-m2-p2) — Mathematics I, chapter 4, asked 2×