The solution of the following partial differential equation $\frac{\partial^{2} u}{\partial x^{2}} = 9 \frac{\partial^{2} u}{\partial y^{2}}$ is

s i n (3 x - y)

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3 x^{2} + y^{2}

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s i n (3 x - 3 y)

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(3 y^{2} - x^{2})

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Solution

The correct option is A $s i n (3 x - y)$
Given $D E$ is
$\frac{\partial^{2} u}{\partial x^{2}} = 9 \frac{\partial^{2} u}{\partial y^{2}}$ ... (i)
Let us verify the options
Option (a): $u = s i n (3 x - y)$
Then $\frac{\partial u}{\partial x} = 3 c o s (3 x - y)$ and $\frac{\partial^{2} u}{\partial x^{2}} = - 9 s i n (3 x - y)$
$\frac{\partial u}{\partial y} = - c o s (3 x - y)$ and $\frac{\partial^{2} u}{\partial y^{2}} = - s i n (3 x - y)$
Hence, option (a) satisfies (1)
So it is the required solution.

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