Question

form the pde by eliminating arbitrary constants and b from z=(x^2+a^2)(y^2+b^2)​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{z=(x^2+a^2)(y^2+b^2)}$

$\underline{\textbf{To find:}}$

$\textsf{A Partial differential equation by eliminating the constants a and b}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{z=(x^2+a^2)(y^2+b^2)}$---------(1)

$\textsf{Differentiate (1) partially with respect to 'x'}$

$\mathsf{\dfrac{{\partial}z}{{\partial}x}=(2x+0)(y^2+b^2)}$

$\mathsf{p=2x(y^2+b^2)}$------(2)

$\textsf{Differentiate (1) partially with respect to 'y'}$

$\mathsf{\dfrac{{\partial}z}{{\partial}y}=(x^2+a^2)(2y+0)}$

$\mathsf{q=2y(x^2+a^2)}$------(3)

$\textsf{Multiplying (1) and (2)}$

$\mathsf{pq=2x(y^2+b^2)\,{\times}\,2y(x^2+a^2)}$

$\mathsf{pq=4xy\,(x^2+a^2)(y^2+b^2)}$

$\mathsf{Using\;(1)}$

$\boxed{\mathsf{pq=4xyz}}$

$\textsf{which is the required PDE}$

$\underline{\textbf{Find more:}}$

Find the second order partial derivatives of F(x.y) = (3x + 2y).4​

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