Question

Form Partial differential eq. of x^2/a^2 +y^2/b^2 +u^2/c^2 =1

Answer 1

Answer on Question #68370 – Math – Differential Equations

Question

Form Partial differential eq. of $x^2/a^2 + y^2/b^2 + u^2/c^2 = 1$

Solution

We have the following equation

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{u^2}{c^2} = 1$$

Note that the number of constants $(a, b, c)$ is more than the number of independent variables $(x, y)$. Hence the order of the resulting differential equation will be more than 1.

Differentiating (1) partially with respect to $x$ we get

$$\frac{2x}{a^2} + \frac{2u}{c^2} \frac{\partial u}{\partial x} = 0$$

or

$$\frac{c^2}{a^2} = -\frac{u}{x} \frac{\partial u}{\partial x}$$

Differentiating (2) partially with respect to $x$ we get

$$\frac{1}{a^2} + \frac{1}{c^2} \frac{\partial u}{\partial x} \frac{\partial u}{\partial x} + \frac{u}{c^2} \frac{\partial^2 u}{\partial x^2} = 0$$

or

$$\frac{c^2}{a^2} = -\left(\frac{\partial u}{\partial x}\right)^2 - u \frac{\partial^2 u}{\partial x^2}$$

From (3) and (4) we get

$$x u \frac{\partial^2 u}{\partial x^2} + x \left(\frac{\partial u}{\partial x}\right)^2 = u \frac{\partial u}{\partial x}$$

which is required differential equation.

This equation is not unique. Similarly differentiating (1) partially with respect to $y$ we get

$$\frac{2y}{b^2} + \frac{2u}{c^2} \frac{\partial u}{\partial y} = 0$$

or

$$\frac{c^2}{b^2} = -\frac{u}{y} \frac{\partial u}{\partial y}$$

and differentiating (6) partially with respect to $y$ we get

$$\frac{1}{b^2} + \frac{1}{c^2} \frac{\partial u}{\partial y} \frac{\partial u}{\partial y} + \frac{u}{c^2} \frac{\partial^2 u}{\partial y^2} = 0$$

or

$$\frac{c^2}{b^2} = -\left(\frac{\partial u}{\partial y}\right)^2 - u \frac{\partial^2 u}{\partial y^2}$$

From (7) and (8) we get

$$y u \frac{\partial^2 u}{\partial y^2} + y \left(\frac{\partial u}{\partial y}\right)^2 = u \frac{\partial u}{\partial y}$$

which is also a required differential equation. Equations (5) and (9) can be summed so that the resulting equation is symmetric with respect to $x$ and $y$

$$x u \frac {\partial^ {2} u}{\partial x ^ {2}} + y u \frac {\partial^ {2} u}{\partial y ^ {2}} + x \left(\frac {\partial u}{\partial x}\right) ^ {2} + y \left(\frac {\partial u}{\partial y}\right) ^ {2} = u \frac {\partial u}{\partial x} + u \frac {\partial u}{\partial y}$$

Answer: The required partial differential equations are

$$x u \frac {\partial^ {2} u}{\partial x ^ {2}} + x \left(\frac {\partial u}{\partial x}\right) ^ {2} = u \frac {\partial u}{\partial x}$$

or

$$y u \frac {\partial^ {2} u}{\partial y ^ {2}} + y \left(\frac {\partial u}{\partial y}\right) ^ {2} = u \frac {\partial u}{\partial y}$$

or

$$x u \frac {\partial^ {2} u}{\partial x ^ {2}} + y u \frac {\partial^ {2} u}{\partial y ^ {2}} + x \left(\frac {\partial u}{\partial x}\right) ^ {2} + y \left(\frac {\partial u}{\partial y}\right) ^ {2} = u \frac {\partial u}{\partial x} + u \frac {\partial u}{\partial y}$$

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