${\left({\mathit{x}}^2+{\mathit{y}}^2\right)}^2$ = xy ... (i)
Differentiating with respect to x, we have,
2 $\left({\mathit{x}}^2+{\mathit{y}}^2\right)\left(2\mathit{x}+2\mathit{y}.\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}\right)$ = y + $\frac{\mathit{x}\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$
⇒ 4x $\left({\mathit{x}}^2+{\mathit{y}}^2\right)$ + 4y $\left({\mathit{x}}^2+{\mathit{y}}^2\right).\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = y + $\frac{\mathit{x}\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}\left(4{\mathit{x}}^2\mathit{y}+4{\mathit{y}}^3-\mathit{x}\right)$ = y - $4{\mathit{x}}^3-4\mathit{x}{\mathit{y}}^2$
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{\mathit{y}-4{\mathit{x}}^3-4\mathit{x}{\mathit{y}}^2}{4{\mathit{x}}^2\mathit{y}+4{\mathit{y}}^3-\mathit{x}}$
y 3cos(logx) 4sin(logx)
Differentiating the above function with respect to x, we have,
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{-3\mathit{s}\mathit{i}\mathit{n}\left(\mathit{l}\mathit{o}\mathit{g}\mathit{x}\right)}{\mathit{x}}$ + $\frac{4\mathit{c}\mathit{o}\mathit{s}\left(\mathit{l}\mathit{o}\mathit{g}\mathit{x}\right)}{\mathit{x}}$
x $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = - 3 sin (logx) + 4 cos (log x)
Again differentiating with respect to x, we have,
x $\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}+\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{-3\mathit{c}\mathit{o}\mathit{s}\left(\mathit{l}\mathit{o}\mathit{g}\mathit{x}\right)}{\mathit{x}}$ - $\frac{4\mathit{s}\mathit{i}\mathit{n}\left(\mathit{l}\mathit{o}\mathit{g}\mathit{x}\right)}{\mathit{x}}$
⇒ ${\mathit{x}}^2\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}+\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + $\mathit{x}\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = - (3 cos (log x) + 4 sin (log x))
⇒ ${\mathit{x}}^2\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}+\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + $\mathit{x}\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = - y
⇒ ${\mathit{x}}^2\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}+\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + $\mathit{x}\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + y = 0