Question

$\frac{d^{2}y}{d{x}^2}+3{\left(\frac{dy}{dx}\right)}^2=x^2\log \left(\frac{d^{2}y}{d{x}^2}\right)$

Answer 1

$y^{\prime \prime }+3{\left(y^{\prime }\right)}^2=x^2\log y^{\prime \prime }$
$\Rightarrow \frac{1}{x^2}(y^{\prime \prime }+3{\left(y^{\prime }\right)}^2)=\log y^{\prime \prime }$
$\Rightarrow \frac{d}{dx}\left(\frac{1}{x^2}(y^{\prime \prime }+3{\left(y^{\prime }\right)}^2)\right)=\frac{1}{y^{\prime \prime }}.y^{\prime \prime \prime }$
$\Rightarrow -\frac{1}{x}(\frac{d^{2}y}{d{x}^2}+3{\left(\frac{dy}{dx}\right)}^2)+\frac{1}{x^2}(\frac{d^{3}y}{d{x}^3}+6\frac{dy}{dx}\frac{d^{2}y}{d{x}^2})=\frac{1}{\frac{d^{2}y}{d{x}^2}}\frac{d^{2}y}{d{x}^2}$
$\Rightarrow \frac{d^{3}y}{d{x}^3}=\frac{d^{2}y}{d{x}^2}(-\frac{1}{x}(\frac{d^{2}y}{d{x}^2}+3{\left(\frac{dy}{dx}\right)}^2))+\frac{1}{x^2}(\frac{d^{3}y}{d{x}^3}+6\frac{dy}{dx}\frac{d^{2}y}{d{x}^2})$