Question

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

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

Answer 1

Answer: (A)

$x\frac{dy}{dx}+y(\ln y)=0\Rightarrow \int \frac{dx}{x}+\int \frac{dy}{y(\ln y)}=C$
$\ln (x\ln y)=C.\text{ If }x=1\text{ then }y=e$
$\Rightarrow \ln (\ln e)=C\Rightarrow C=0\Rightarrow (A)]$