Question

The type of the partial differential equation $\frac{\partial f}{\partial t}=\frac{{\partial }^{2}f}{\partial x^2}$ is

• A. Parabolic
• B. Elliptic
• C. Hyperbolic
• D. Nonlinear

Answer 1

The correct option is A Parabolic

A $2nd$ order partial differential equation of the form $A\frac{{\partial }^{2}u}{\partial x^2}+B\frac{{\partial }^{2}u}{\partial x\partial y}+C\frac{{\partial }^{2}u}{\partial y^2}+f(x,y,u,p,q)=0$ will be
(i) Parabolic if $B^2-4AC=0$
(ii) Hyperbolic if $B^2-4AC>0$
(iii) Elliptic if $B^2-4AC<0$
Given equation can be written as
$1.\frac{{\partial }^{2}f}{\partial x^2}+0.\frac{{\partial }^{2}f}{\partial x\partial t}+0.\frac{{\partial }^{2}f}{\partial t^2}+(-\frac{\partial f}{\partial t})=0$
On comparison, $A=1,B=0,C=0$
Now, $B^2-4AC=0-4\left(1\right)\left(0\right)=0$
$\therefore$ The given equation is parabolic.