Question

$\begin{gathered} \sf{If \: u = x\Phi( \frac{y}{x} ) + \Psi( \frac{y}{x} ) \: then \: find} \\ \sf{x}^{2} \frac{ \partial^{2} u}{ \partial {x}^{2} } + 2xy \frac{ \partial {}^{2} u}{ \partial x \cdot \partial y} + {y}^{2} { \frac{\partial^{2} u}{\partial {y}^{2} } } \end{gathered}$​

Answer 1

Answer:

then the complement of it = 90-x . given an angle is equal to 5 times it's it's complement. x= 75. hence the measure of an angel=7

Answer 2

$\sf \: u = x\Phi( \frac{y}{x} ) + \Psi( \frac{y}{x})$

$\\ \implies \sf \frac{∂ ^{2}u }{∂ {x}^{2} } = \frac{∂ ^{2} }{∂ {x}^{2} } [ \sf \: x\Phi (\frac{y}{x} ) + \Psi( \frac{y}{x})]$

$\implies \sf \frac{∂ ^{2}u }{∂ {x}^{2} } = 0 + 2 \Psi y \frac{1}{ {x}^{3} }$

$\implies \sf \frac{∂ ^{2}u }{∂ {x}^{2} } = \frac{ 2 \Psi y}{ {x}^{3} } \: \: ...(1)$

$\\ \implies \sf\frac{ \partial {}^{2} u}{ \partial x \cdot \partial y} = \frac{ \partial }{ \partial x} (\frac{ \partial u }{ \partial y} )$

$\implies \sf\frac{ \partial {}^{2} u}{ \partial x \cdot \partial y} = \frac{ \partial }{ \partial x} (\frac{ \partial u }{ \partial y} )$

$\implies \sf\frac{ \partial {}^{2} u}{ \partial x \cdot \partial y} = \frac{ \partial }{ \partial x} [ \frac{ \partial }{ \partial y}(\sf \: x\Phi (\frac{y}{x} ) + \Psi( \frac{y}{x})) ]$

$\implies \sf\frac{ \partial {}^{2} u}{ \partial x \cdot \partial y} = \frac{ \partial }{ \partial x} [ \Phi + \frac{ \Psi}{x} ]$

$\implies \sf\frac{ \partial {}^{2} u}{ \partial x \cdot \partial y} = 0 - \frac{ \Psi}{x ^{2} }$

$\implies \sf\frac{ \partial {}^{2} u}{ \partial x \cdot \partial y} = - \frac{ \Psi}{x ^{2} } \: \: ...(2)$

$\\ \implies \sf \frac{∂ ^{2}u }{∂ {y}^{2} } = \frac{∂ ^{2} }{∂ {y}^{2} } [ \sf \: x\Phi (\frac{y}{x} ) + \Psi( \frac{y}{x})]$

$\implies \sf \frac{∂ ^{2}u }{∂ {y}^{2} } = 0 + 0$

$\implies \sf \frac{∂ ^{2}u }{∂ {y}^{2} } = 0 \: \: ...(3)$

$\sf \purple{from \: \: (1) \: (2) \: and \: (3)}$

$\sf{x}^{2} \frac{ \partial^{2} u}{ \partial {x}^{2} } + 2xy \frac{ \partial {}^{2} u}{ \partial x \cdot \partial y} + {y}^{2} { \frac{\partial^{2} u}{\partial {y}^{2}}}$

$\implies\sf{x}^{2} (\frac{ 2 \Psi y}{ {x}^{3} } ) + 2xy (- \frac{ \Psi}{x ^{2} }) + {y}^{2} (0)$

$\implies\sf \frac{ 2 \Psi y}{ {x} } - \frac{ 2\Psi y}{x } +0$

$\therefore\sf 0$

$\orange {\therefore \sf{x}^{2} \frac{ \partial^{2} u}{ \partial {x}^{2} } + 2xy \frac{ \partial {}^{2} u}{ \partial x \cdot \partial y} + {y}^{2} { \frac{\partial^{2} u}{\partial {y}^{2}}} = 0}$

HOPE IT HELPS!!