The solution curve of the differential equation [y (d x)/(d y)=x( _e x- _e y+1), x0, y0]…

Question

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

Accepted Answer

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

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

$$\begin{aligned}
& \frac{d x}{d y}=t+y \frac{d t}{d y} \
& t+y \frac{d t}{d y}=t(\ln (t)+1)
\end{aligned}$$

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

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

$$\Rightarrow \int \frac{d p}{p}=\int \frac{d y}{y} \quad$$ let $$\ln t=p$$

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

$$\begin{aligned}
& \Rightarrow \ln p=\ln y+c \
& \ln (\ln t)=\ln y+c \
& \ln \left(\ln \left(\frac{x}{y}\right)\right)=\ln y+c \
& \text { at } x=e, y=1 \
& \ln \left(\ln \left(\frac{e}{1}\right)\right)=\ln (1)+c \Rightarrow c=0
\end{aligned}$$

$$\begin{aligned}
& \ln \left|\ln \left(\frac{x}{y}\right)\right|=\ln y \
& \left|\ln \left(\frac{x}{y}\right)\right|=e^{\ln y} \
& \left|\ln \left(\frac{x}{y}\right)\right|=y
\end{aligned}$$

Solutions · 1