Consider the following equations [begingathered V( x, y ) x = px^2 + y^2 + 2xy hfill V( x, y ) y = x^2 + qy^2 + 2xy hfill endgathered ] where p and q are constants. V(x, y) that satisfies the above equations is

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Calculus

Consider the following equations
\[\begin{gathered} \frac{{\partial {\text{V}}\left( {{\text{x, y}}} \right)}}{{\partial {\text{x}}}} = {\text{p}}{{\text{x}}^2} + {{\text{y}}^2} + 2{\text{xy}} \hfill \\ \frac{{\partial {\text{V}}\left( {{\text{x, y}}} \right)}}{{\partial {\text{y}}}} = {{\text{x}}^2} + {\text{q}}{{\text{y}}^2} + 2{\text{xy}} \hfill \\ \end{gathered} \]
where p and q are constants. V(x, y) that satisfies the above equations is

A) \[{\text{p}}\frac{{{{\text{x}}^3}}}{3} + {\text{q}}\frac{{{{\text{y}}^3}}}{3} + 2{\text{xy}} + 6\] B) \[{\text{p}}\frac{{{{\text{x}}^3}}}{3} + {\text{q}}\frac{{{{\text{y}}^3}}}{3} + 5\] C) \[{\text{p}}\frac{{{{\text{x}}^3}}}{3} + {\text{q}}\frac{{{{\text{y}}^3}}}{3} + {{\text{x}}^2}{\text{y}} + {\text{x}}{{\text{y}}^2} + {\text{xy}}\] D) \[{\text{p}}\frac{{{{\text{x}}^3}}}{3} + {\text{q}}\frac{{{{\text{y}}^3}}}{3} + {{\text{x}}^2}{\text{y}} + {\text{x}}{{\text{y}}^2}\]

Answer & Solution

Correct Answer: Option D

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