The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial x \partial y } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = u$ is Select all that apply.
A) linear
B) nonlinear
C) hyperbolic
D) elliptic
E) parabolic

Question

Quizplus · Accepted Answer

The given differential equation is linear because it involves linear operations on the function $u$ and its derivatives (no terms involve powers or products of $u$ or its derivatives). It is elliptic because the discriminant of the characteristic equation derived from the highest order partial derivatives, $B^2 - 4AC$, where $A = 1$, $B = 1$, and $C = 1$, is negative ($1^2 - 4(1)(1) = -3$).

The Differential Equation Is Select All That Apply