Question

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

Answer 1

$z=x+ax^2y^2....................(1)$

From the equation above, we have that the number of arbitrary constants is less than the number of independent variables. This implies that we will have a multiple PDE with order of one.

Differentiate (1) w.r.t $x$ and $y$

$\frac{\partial z}{\partial x}=p=1+2axy^2, \frac{\partial z}{\partial y}=q=2ax^2y.............(2)$

From (2), we have that

$a=\frac{p-1}{2xy^2},a=\frac{q}{2x^2y}$

Put each values of a into (1). We have that;

$z=x+(\frac{p-1}{2xy^2})\cdot x^2y^2\\ z=x+(\frac{p-1}{2})x\\ z=x(\frac{p+1}{2})\\ \text{Also, }\\ z=x+(\frac{q}{2x^2y})\cdot x^2y^2\\ z=x+\frac{q}{2x}y\\ 2xz=2x^2+qy.$

So, the two partial differential equations for from the equation above are $z=x(\frac{p+1}{2})$ and $2xz=2x^2+qy.$

We can see the the two PDEs are of order one and degree one. The independent variables are x and y. While the dependent variable is z.