Question

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

Answer 1

Answer on Question #68367 – Math – Differential Equations

Question

Form Partial differential eq. of

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

Solution

We have a relation

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

We differentiate (1) first with respect to $x$ and then with respect to $y$

$$2 \frac {\partial u}{\partial x} = \frac {2 x}{a ^ {2}} \quad \text {or} \quad \frac {1}{a ^ {2}} = \frac {1}{x} \frac {\partial u}{\partial x} = \frac {p}{x}$$ $$2 \frac {\partial u}{\partial y} = \frac {2 y}{b ^ {2}} \quad \text {or} \quad \frac {1}{b ^ {2}} = \frac {1}{y} \frac {\partial u}{\partial y} = \frac {q}{y}$$

Substituting these values of $1 / a^2$ and $1 / b^2$ into (1) we get

$$2 u = x p + y q$$

or

$$2 u = x \frac {\partial u}{\partial x} + y \frac {\partial u}{\partial y}.$$

Answer: The required partial equation is $2u = xp + yq$ , that is, $2u = x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}$ .

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