---
title: "Classifying Second-Order PDEs and What the Type Decides"
description: "The sign pattern of the second-order coefficients sorts equations into elliptic, parabolic and hyperbolic. The type decides which data may be prescribed and whether a solution smooths its data out or "
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source: https://lightmysky.com/learn/mathematics/classifying-second-order-pdes-and-what-the-type-decides-mt_hlQWbd1kzR.md
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---

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# Classifying Second-Order PDEs and What the Type Decides

The sign pattern of the second-order coefficients sorts equations into elliptic, parabolic and hyperbolic. The type decides which data may be prescribed and whether a solution smooths its data out or carries it along.

Subject: Mathematics · Area: Differential Equations · Ages 22 to 23
Page: https://lightmysky.com/learn/mathematics/classifying-second-order-pdes-and-what-the-type-decides-mt_hlQWbd1kzR

## Ready when they can

- Classify a given second-order equation from its coefficients
- Match each type with a set of boundary or initial conditions that makes the problem well posed
- Say which type smooths a kink in the data and which one keeps it

## Lesson: Sorting second-order equations by their type

Heat in a rod, a humming string, a steady voltage: one short test sorts them. Look only at the second-order part, written as A times u_xx plus B times u_xy plus C times u_yy. Form the discriminant B squared minus 4AC, a shape you already know from quadratics. A negative value means elliptic, zero means parabolic, and positive means hyperbolic. First-order terms, plain u terms, and the right hand side never enter this test.

**Example.** Try three classics. u_xx plus 4u_yy equals 0 has A 1, B 0, C 4: 0 minus 16 is negative, so elliptic. The heat case u_t equals 5u_xx has only u_xx at second order: 0, so parabolic. The wave case u_tt minus 9u_xx equals 0 has A 1, B 0, C negative 9: 36, so hyperbolic.

Each type needs its own data. Elliptic means steady state, so give u around the whole closed boundary. Parabolic carries an initial profile forward, so give u at time zero. Hyperbolic is second order in time, so give the starting position and how fast each part moves.

The type also decides the fate of a sharp kink. Parabolic flow averages each point with its neighbors, so a kink smooths out right away. Hyperbolic motion carries every feature along its characteristics, so a kink travels unchanged.

**Recap.** The sign of B squared minus 4AC names the type, and the type decides the data and the fate of kinks.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Symmetric Matrices and the Spectral Theorem](https://lightmysky.com/learn/mathematics/symmetric-matrices-and-the-spectral-theorem-mt_sdQv4m7Nbk)
- [Partial Derivatives](https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9)
- [Phase Portraits, Equilibria and Stability](https://lightmysky.com/learn/mathematics/phase-portraits-equilibria-and-stability-mt_ULuZk4lmGr)

## Opens up

- [Separation of Variables and the Heat Equation](https://lightmysky.com/learn/mathematics/separation-of-variables-and-the-heat-equation-mt_XnPDL0X1TL)