---
title: "Solve a partial differential equation"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:solve-a-partial-differential-equation"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-a-partial-differential-equation
site: BCT Engineering
---

# Solve a partial differential equation

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Chaitra (new course), 2 marks
- Video: [Oxford Calculus: Solving Simple PDEs](https://www.youtube.com/watch?v=uztjxrGY6Jw) — Tom Rocks Maths
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-a-partial-differential-equation
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Solve the partial differential equation ∂²z/∂x∂y = sin x sin y given that ∂z/∂y = −2 sin y when x = 0, and z = 0 when y = π/2.

## Short answer

z = (1+cos x) cos y. Integrating ∂²z/∂x∂y gives ∂z/∂y = −(1+cos x) sin y; condition yields no extra term, then integration in y gives solution.

## Answer

Integrate the equation twice, once with respect to each variable, and use the conditions to find the arbitrary functions.

```text
∂²z/∂x∂y = sin x sin y
Integrate w.r.t. x:   ∂z/∂y = −cos x sin y + g(y)
Use x = 0, ∂z/∂y = −2 sin y:
                      −2 sin y = −sin y + g(y)   so  g(y) = −sin y
Hence                 ∂z/∂y = −(1 + cos x) sin y
Integrate w.r.t. y:   z = (1 + cos x) cos y + h(x)
Use y = π/2, z = 0:   0 = 0 + h(x)   so  h(x) = 0
```

**Answer:** z = (1 + cos x) cos y

Check: ∂z/∂x = −sin x cos y, so ∂²z/∂x∂y = sin x sin y.

## Check yourself

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

1. The PDE is solved by:
   A. integrating once in x, then in y, adding arbitrary functions
   B. separation of variables
   C. Lagrange's method
   D. Fourier series
   Answer: A. Direct integration. (Key point: Method)

2. From x = 0 and ∂z/∂y = −2 sin y, g(y) equals:
   A. −2 sin y
   B. −sin y
   C. sin y
   D. 0
   Answer: B. −2 sin y = −sin y + g. (Key point: g(y))

3. z =
   A. (1 − cos x) cos y
   B. (1 + cos x) sin y
   C. (1 + cos x) cos y
   D. cos x cos y
   Answer: C. h(x) = 0 from z = 0 at y = π/2. (Key point: Result)

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