The solution curve of the differential equation y(dx)/(dy)=x( _ex- _ey+1),x0,y0 passing …

Question

The solution curve of the differential equation 
$y\frac{dx}{dy}=x({\log }_{e}x-{\log }_{e}y+1),x>0,y>0$ passing through the point $(e,1)$ is

Accepted Answer

$\frac{dx}{dy}=\frac{x}{y}(\ln (\frac{x}{y})+1)$

Let $\frac{x}{y}=t\Rightarrow x=ty$

$\begin{gathered} \frac{dx}{dy}=t+y\frac{dt}{dy} \ t+y\frac{dt}{dy}=t(\ln (t)+1) \end{gathered}$

$y\frac{dt}{dy}=t\ln (t)\Rightarrow \frac{dt}{t\ln (t)}=\frac{dy}{y}$

$\Rightarrow \int \frac{dt}{t\cdot \ln (t)}=\int \frac{dy}{y}$

$\Rightarrow \int \frac{dp}{p}=\int \frac{dy}{y}\,$ let $\ln t=p$

$\frac{1}{t}dt=dp$

$\begin{gathered} \Rightarrow \ln p=\ln y+c \ \ln (\ln t)=\ln y+c \ \ln (\ln (\frac{x}{y}))=\ln y+c \ x=e,y=1 \ \ln (\ln (\frac{e}{1}))=\ln (1)+c\Rightarrow c=0 \end{gathered}$

$\begin{gathered} \ln |\ln (\frac{x}{y})|=\ln y \ |\ln (\frac{x}{y})|={e}^{\ln y} \ |\ln (\frac{x}{y})|=y \end{gathered}$

Solutions · 1