Question

By eliminating a & b from z = ax + by + (a/b) then, P.D.E formed is _____

• A. \( z = px + qy + (p/q) \)
• B. \( z = px + qy + \log(pq) \)
• C. \( z = ax + by + (a/b) \)
  % Original equation repeated
• D. \( z = ax + by + \log(ab) \)

Hint:

Forming PDEs by Eliminating Constants. If an equation involves arbitrary constants (like a, b), find partial derivatives \(p = \partial z / \partial x\) and \(q = \partial z / \partial y\). Use these derivative equations along with the original equation to eliminate the constants.

Answer 1

The Correct Option is A

We are given the relation \( z = ax + by + (a/b) \), where a and b are arbitrary constants.
We need to form a partial differential equation (PDE) by eliminating a and b.
We use the standard notation for partial derivatives of z with respect to x and y: $$ p = \frac{\partial z}{\partial x} $$ $$ q = \frac{\partial z}{\partial y} $$ Differentiate the given relation partially with respect to x (treating y, a, b as constants): $$ \frac{\partial z}{\partial x} = a $$ So, \( p = a \).
Differentiate the given relation partially with respect to y (treating x, a, b as constants): $$ \frac{\partial z}{\partial y} = b $$ So, \( q = b \).
Now substitute \(a=p\) and \(b=q\) back into the original relation: $$ z = (p)x + (q)y + \frac{p}{q} $$ $$ z = px + qy + \frac{p}{q} $$ This is the required PDE formed by eliminating the arbitrary constants a and b.
This matches option (1).
This form is related to Clairaut's partial differential equation.