Question

eliminate the arbitrary constant a and b from z=ax+by+a2b2, to obtain the partial differential equation.​

Answer 1

z=ax+by+a^2+b^2 differential

Answer 2

The required partial differential equation is z = px + qy + p²q²

Given :

The given equation z = ax + by + a²b²

To find :

The partial differential equation

Solution :

Step 1 of 3 :

Write down the given equation

Here the given equation is

z = ax + by + a²b² - - - - - - (1)

Step 2 of 3 :

Find the partial derivatives

z = ax + by + a²b²

Differentiating both sides partially with respect to x we get

$\displaystyle \sf{ \frac{ \partial z}{ \partial x} = a}$

$\displaystyle \sf{ \implies p = a}$

Differentiating both sides partially with respect to y we get

$\displaystyle \sf{ \frac{ \partial z}{ \partial y} = b}$

$\displaystyle \sf{ \implies q = b}$

Step 3 of 3 :

Form the partial differential equation

Putting the value of a and b in Equation 1 we get

z = px + qy + p²q²

Which is the required partial differential equation