Partial Differential Equations

MULTIPLE-CHOICE QUESTIONS

1 In a general second-order linear partial differential equation with two independent variables

$A\frac{{\partial }^{2}u}{\partial x^2}+B\frac{{\partial }^{2}u}{\partial x\partial y}+C\frac{{\partial }^{2}u}{\partial y^2}+D=0$

where $A$, $B$, $C$ are functions of $x$ and $y$, and $D$ is a function of $x$, $y$, $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$, then the partial differential equation is parabolic if

(A) $B^2-4AC<0$ (B) $B^2-4AC>0$ (C) $B^2-4AC=0$(D) $B^2-4AC\ne 0$

Answer: (C) $B^2-4AC=0$.

2 A partial differential equation requires

(A) exactly one independent variable (B) two or more independent variables

(C) more than one dependent variable (D) equal number of dependent and independent variables

Answer: (B) two or more independent variables.

3 Using substitution, which of the following equations are solutions to the partial differential equation?

$\frac{{\partial }^{2}u}{\partial x^2}=9\frac{{\partial }^{2}u}{\partial y^2}$

(A)