Question

solve the partial differential equation by eliminating arbitrary constant z=(x+a)(y+b)​

Answer 1

Answer:

z = pq

Step-by-step explanation:

z = (x+a) (x+b)

z = xy + ax + by + ab

$p = \frac{\partial z}{\partial x} = y + b\\q = \frac{\partial z}{\partial y} = x +b$

z = pq

Answer 2

The required partial differential equation is z = pq

Given :

The equation z = ( x + a ) ( y + b )

To find :

The partial differential equation by eliminating arbitrary constant

Solution :

Step 1 of 2 :

Write down the given equation

The given equation is

$\displaystyle \sf{ } z = ( x + a ) ( y + b )$

Step 2 of 2 :

Form the partial differential equation

$\displaystyle \sf{ } z = ( x + a ) ( y + b ) \: \: \: \: - - - (1)$

Differentiating both sides of Equation 1 partially with respect to x we get

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

$\displaystyle \sf{ \implies p = y + b \: \: \: \: - - - (2)}$

Differentiating both sides of Equation 1 partially with respect to y we get

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

$\displaystyle \sf{ \implies q = x + a \: \: \: \: - - - (3)}$

Using Equation 2 and Equation 3 From Equation 1 we get

$\displaystyle \sf{ \implies z = qp }$

$\displaystyle \sf{ \implies z = p q}$

Which is the required partial differential equation

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