If the solution curve of the differential equation [(d y)/(d x)=(x+y-2)/(x-y)] passes thro…

Question

If the solution curve of the differential equation $$\frac{d y}{d x}=\frac{x+y-2}{x-y}$$ passes through the points $(2,1)$ and $$(\mathrm{k}+1,2), \mathrm{k}>0$$, then

Accepted Answer

$\frac{d y}{d x}=\frac{x+y-2}{x-y}=\frac{(x-1)+(y-1)}{(x-1)-(y-1)}$

Let $x-1=X, y-1=Y$

$$
\frac{d Y}{d X}=\frac{X+Y}{X-Y}
$$

Let $Y=t X \Rightarrow \frac{d Y}{d X}=t+X \frac{d t}{d X}$

$t+X \frac{d t}{d X}=\frac{1+t}{1-t}$

$X \frac{d t}{d X}=\frac{1+t}{1-t}-t=\frac{1+t^{2}}{1-t}$

$\int \frac{1-t}{1+t^{2}} d t=\int \frac{d X}{X}$

$$
\begin{aligned}
&\tan ^{-1} t-\frac{1}{2} \ln \left(1+t^{2}\right)=\ln |X|+c \\
&\tan ^{-1}\left(\frac{y-1}{x-1}\right)-\frac{1}{2} \ln \left(1+\left(\frac{y-1}{x-1}\right)^{2}\right)=\ln |x-1|+c
\end{aligned}
$$

Curve passes through $(2,1)$

$0-0=0+c \Rightarrow c=0$

If $(k+1,2)$ also satisfies the curve

$$
\begin{aligned}
&\tan ^{-1}\left(\frac{1}{k}\right)-\frac{1}{2} \ln \left(\frac{1+k^{2}}{k^{2}}\right)=\ln k \\
&2 \tan ^{-1}\left(\frac{1}{k}\right)=\ln \left(1+k^{2}\right)
\end{aligned}
$$

Solutions · 1