Question

form the partial differential equation by eliminating a and b from z = a x + b y​

Answer 1

Given,

$\longrightarrow z=ax+by\quad\quad\dots(1)$

Differentiating $z$ partially wrt $x,$

$\longrightarrow\dfrac{\partial z}{\partial x}=a$

Differentiating $z$ partially wrt $y,$

$\longrightarrow\dfrac{\partial z}{\partial y}=b$

By replacing $a$ and $b$ by these in (1), we're finding the required partial differential equation by eliminating $a$ and $b.$

Then, replacing $a$ and $b$ in (1) we get,

$\longrightarrow\underline{\underline{z=\dfrac{\partial z}{\partial x}\,x+\dfrac{\partial z}{\partial y}\,y}}$

This is the required differential equation.

Take,

• $p=\dfrac{\partial z}{\partial x}$

• $q=\dfrac{\partial z}{\partial y}$

Then,

$\longrightarrow\underline{\underline{z=px+qy}}$

The partial differential equation can be written in this form also.

Answer 2

equation is z = ax + by + ab

here it is clearly shown that a and b are two arbitrary constant.

differentiating given equation partially with respect to x,

z = ax + by + ab

differentiating the above function partially w.r.t x

∂z/∂x = a

i.e., p = a  (1)

differentiating the above function partially w.r.t y

∂z/∂y = b

i.e., q = b  (2)

from (1) and (2), we get,

a = p and b = q

substituting the  values of a and b in given equation, we get,

z = ax + by + ab

z = ax + by + (p) (q)

∴ z = ax + by + pq