Question

Q. Obtain the partial differential equation by eleminating the arbitrary constant from the relation
u=xy+y√(x^2-a^2 )+b

Answer 1

Answer on Question #68401 – Math – Differential Equations

Question

Obtain the partial differential equation by eliminating the arbitrary constant from the relation

$$u = xy + y\sqrt{x^2 - a^2} + b$$

Solution

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

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

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