Question

Q2. (a) For each of the following, investigate whether the partial differential equation Is linear, quasi-linear or nonlinear. If it is linear, state whether it is homogeneous or non

i) uxxxx +2uxxyy + Uyyy= 0

ii) uxx +2uxy + uyy= sin x

iii) uxx +xuy = y 

Q2. (b) Obtain PDE z = x + ax2y2 where a, b are arbitrary constants and also analyze the partial differential equation. 

Q2. (c) Find the general solution of the first-order linear partial differential equation xux + yuy =u it and investigate first-order, quasi-linear and linear partial differential equations. 

Answer 1

a) A partial differential equation is said to be linear if it is linear with respect to the unknown function and its derivatives that appear in it.

A partial differential equation is said to be quasilinear if it is linear with respect to all the highest order derivatives of the unknown function.

i) Linear, homogeneous

ii) Linear, non-homogeneous

iii) Linear, non-homogeneous

b)

$\frac{\partial z}{\partial x}=1+2axy^2,\frac{\partial^2 z}{\partial x^2}=2ay^2$

$\frac{\partial z}{\partial y}=2ayx^2,\frac{\partial^2 z}{\partial y^2}=2ax^2$

$\frac{\partial^2 z}{\partial x^2}+\frac{\partial^2 z}{\partial y^2}=2a(x^2+y^2)$

Linear, non-homogeneous equation.

c) This is first-order linear partial differential equation.

$\frac{dx}{x}=\frac{dy}{y}=\frac{du}{u}$

$lnx=lny+lnc_1$

$\frac{x}{y}=c_1$

$lny=lnu+lnc_2$

$\frac{y}{u}=c_2$