Question

If the curve satisfying $ydx+(x+x^{2}y)dy=0$ passes through (1,1), then $4\log y\left(4\right)-\frac{1}{y\left(4\right)}$ is equal to

• A. 4
• B. 3
• C. 16
• D. 8

Answer 1

The correct option is A 4

The presence of $ydx+xdy$ suggest a factor of the form $f\left(xy\right)$

$ydx+xdy+x^{2}dy=0\Rightarrow \frac{ydx+xdy}{x^2{y}^2}+\frac{dy}{y}=0$

$\Rightarrow \frac{d\left(xy\right)}{{\left(xy\right)}^2}+\frac{dy}{y}=0$

$\Rightarrow -\frac{1}{xy}+\log y=c$

For $y\left(1\right)=1\Rightarrow c=-1$

Then $4\log y\left(4\right)-\frac{1}{y\left(4\right)}=4$