---
title: "Classify a second order PDE"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:classify-a-second-order-pde"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/classify-a-second-order-pde
site: BCT Engineering
---

# Classify a second order PDE

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Chaitra (new course), 2 marks
- Video: [Classification of Second-Order PDE | Partial Differential Equation Example  & Concepts  by GP Sir](https://www.youtube.com/watch?v=2DmqG66fVhg) — Dr.Gajendra Purohit
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/classify-a-second-order-pde
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Write down the classification rules for a general second order partial differential equation and use them to classify the equation u_xx + x²u_yy = 0.

## Short answer

For u_xx + x² u_yy = 0, discriminant B²−4AC = −4x². Hence elliptic where x≠0 (negative), and parabolic on the line x=0 (zero).

## Answer

A general second order PDE in two variables is A u_xx + B u_xy + C u_yy + (lower order terms) = 0. Its type depends on the sign of the discriminant B² − 4AC at the point considered:

| Condition | Type | Example |
|---|---|---|
| B² − 4AC < 0 | Elliptic | Laplace equation |
| B² − 4AC = 0 | Parabolic | Heat equation |
| B² − 4AC > 0 | Hyperbolic | Wave equation |

**Given:** u_xx + x² u_yy = 0. Here A = 1, B = 0, C = x².

B² − 4AC = 0 − 4x² = −4x²

- For x ≠ 0, B² − 4AC < 0, so the equation is **elliptic**.
- On the line x = 0, B² − 4AC = 0, so it is parabolic there (the equation degenerates to u_xx = 0).

**Answer:** elliptic everywhere except on the y-axis (x = 0), where it is parabolic.

## Check yourself

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

1. B^2 − 4AC < 0 means:
   A. hyperbolic
   B. parabolic
   C. linear
   D. elliptic
   Answer: D. Standard classification. (Key point: Rule)

2. For u_xx + x^2 u_yy = 0:
   A. A = 1, B = 0, C = x^2
   B. A = x^2, B = 0, C = 1
   C. A = 1, B = x, C = 1
   D. A = 1, B = 1, C = x^2
   Answer: A. Read from principal part. (Key point: Coefficients)

3. The equation is:
   A. hyperbolic everywhere
   B. elliptic except on x = 0 where parabolic
   C. elliptic everywhere
   D. parabolic everywhere
   Answer: B. B^2 − 4AC = −4x^2. (Key point: Result)

## Related in Mathematics III

- [Classify a second order PDE](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aclassify-a-second-order-pde-two) — 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) — 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-three) — Mathematics III, chapter 4

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