Question

The solution to the differential equation $y\ln y+x{y}^{\prime }=0,$ where $y\left(1\right)=e,$ is

• A. $|x\ln y|=1$
• B. $xy\left(\ln y\right)=1$
• C. ${\left(\ln y\right)}^2=2$
• D. $\ln y+\left(\frac{x^2}{2}\right)y=1$

Answer 1

The correct option is A $|x\ln y|=1$

$x\frac{dy}{dx}+y\left(\ln y\right)=0$
$\Rightarrow \int \frac{dx}{x}+\int \frac{dy}{y\left(\ln y\right)}=0$
$\Rightarrow \ln |x|+\ln |\left(\ln y\right)|=\ln |c|$
$\Rightarrow |x\ln y|=|c|$
$y\left(1\right)=e\Rightarrow c=\pm 1$
Hence, the equation of the curve is $|x\ln y|=1$