Question

Let $y=y\left(x\right)$ be a curve satisfying the differential equation $y\left(\frac{d^{2}y}{d{x}^2}\right)=2{\left(\frac{dy}{dx}\right)}^2.$ If the curve passes through $(2,2)$ and $(8,\frac{1}{2}),$ then the value of $y\left(\frac{1}{9}\right)$ is

Answer 1

Given, $y\left(\frac{d^{2}y}{d{x}^2}\right)=2{\left(\frac{dy}{dx}\right)}^2$
$\Rightarrow \frac{y^{\prime \prime }}{y^{\prime }}=\frac{2y^{\prime }}{y}$
Integrating both sides,
$\ln y^{\prime }=2\ln y+\ln a$
$\Rightarrow \frac{y^{\prime }}{y^2}=a$
$\Rightarrow \int \frac{1}{y^2}dy=\int adx$
$\Rightarrow \frac{-1}{y}=ax+b$
Since the curve passes through $(2,2)$ and $(8,\frac{1}{2})$, so
$2a+b=\frac{-1}{2}$
$8a+b=-2$
Solving the above equations, we get $a=\frac{-1}{4},b=0$
Hence, $y=\frac{4}{x}$
$y\left(\frac{1}{9}\right)=36$