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) u xxx +2uxxyy + uyyy= 0

ii) . u,x +2Uxy Uyy= sin x

iii) .uxx +xuy = y 

Answer 1

Solution:

(i) This PDE is linear as the dependent variable and all its partial derivatives occur linearly here. Next, all the terms of this PDE contain the dependent variable's partial derivatives, so it is homogeneous.

(ii) This PDE is linear but non-homogeneous as there is just 'sin x' on right side which is without u or its partial derivative.

(iii) This PDE is linear but non-homogeneous as there is just 'y' on right side which is without u or its partial derivative.