Question

Form Partial differential eq. of u=xy+y√(x^2-a^2 )+b

Answer 1

Answer on Question #68368 – Math – Differential Equations

Question

Form Partial differential eq. of $u = xy + y\sqrt{(x^2 - a^2)} + b$

Solution

$$\begin{array}{l} u = x y + y \sqrt {x ^ {2} - a ^ {2}} + b \\ p = \frac {\partial u}{\partial x} = y + \frac {x y}{\sqrt {x ^ {2} - a ^ {2}}} \\ q = \frac {\partial u}{\partial y} = x + \sqrt {x ^ {2} - a ^ {2}} \\ \sqrt {x ^ {2} - a ^ {2}} = q - x \rightarrow p = y + \frac {x y}{q - x} \\ \end{array}$$

A partial differential equation (PDE) is

$$p = y + \frac {x y}{q - x},$$

that is,

$$\frac {\partial u}{\partial x} = y + \frac {x y}{\frac {\partial u}{\partial y} - x}$$

Answer: $p = y + \frac{xy}{q - x}$ or $\frac{\partial u}{\partial x} = y + \frac{xy}{\frac{\partial u}{\partial y} - x}$.

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