Question

The family of integral curves of the differential equation $\frac{dy}{dx}+x^{3}y=x$ is cut by the line $x=2;$ the tangents at the points of intersection are concurrent at $(\lambda ,\mu )$. Then find the value of $\left[\frac{\lambda }{\mu }\right]$, where $[.]$ denotes greatest integer function.

Answer 1

Answer: 8

$\frac{dy}{dx}+x^{3}y=x$
Equation of tangent at $(2,\alpha )$
$(y-\alpha )=(2-8\alpha )(x-2)$
$\Rightarrow y-\alpha =2x-4-8\alpha (x-2)$
$\Rightarrow (y-2x+4)+\alpha (8x-17)=0$
$x=\frac{17}{8}\text{ and }y=2x-4$
$\Rightarrow y=\frac{1}{4}$
$\therefore (\lambda ,\mu )\equiv \left(\frac{17}{8},\frac{1}{4}\right)$
$\therefore \frac{\lambda }{\mu }=\frac{17}{2}$
$\Rightarrow \left[\frac{\lambda }{\mu }\right]=8$