$\text{ Solve }\frac{dy}{dx}=\frac{3y-7x+7}{3x-7y-3}\text{. }$

$\frac{dy}{dx}=\frac{3y-7x+7}{3x-7y-3}\begin{array}{r}a=-7,b=3\\ a^{\mathrm{\prime }}=3,b^{\mathrm{\prime }}=-7\end{array}\frac{a}{a},\ne \frac{b}{b}$

$\text{ Put }x=X+h,y=Y+k\text{ then }\frac{dy}{dx}=\frac{dY}{dX}$

$\frac{dY}{dX}=\frac{3(Y+k)-7(X+h)+7}{3(X+h)-7(Y+k)-3}$

$=\frac{3Y-7X+(3k-7h+7)}{3X-7Y+(3h-7k-3)}......\left(1\right)$

$\text{ Put }-7h+3k+7=0.....\left(2\right)$

$3h-7k-3=0........\left(3\right)$

Solving (2)& (3) we get h,k

$\begin{array}{cccc}h& k& 1& \\ 3& 7& -7& 3\\ -7& -3& 3& -7\end{array}$

$\Rightarrow \frac{h}{-9+49}=\frac{k}{21-21}=\frac{1}{49-9}$

$\Rightarrow \frac{h}{40}=\frac{k}{0}=\frac{1}{40}\Rightarrow h=1;k=0$

$\text{ Put }Y=VX\Rightarrow \frac{dY}{dX}=v+X\frac{dV}{dX}$

$\text{ From (1) and (2) }V+X\frac{dV}{dX}=\frac{3V-7}{3-7V}$

$X\frac{dV}{dX}=\frac{3V-7}{3-7V}-\frac{V}{1}=\frac{3V-7-3V+7V^2}{3-7V}$

$X\frac{dV}{dX}=\frac{7V^2-7}{3-7V}\Rightarrow \frac{3-7V}{7V^2-7}dV=\frac{dX}{X}$

$\Rightarrow -\int \frac{7V-3}{7\left(V^2-1\right)}dV=\int \frac{dX}{X}$

$\Rightarrow \int \frac{7V-3}{(V-1)(V+1)}=-7\int \frac{dX}{X}$

$\Rightarrow \int \left[\frac{2}{V-1}+\frac{5}{V+1}\right]dV=-7\int \frac{dX}{X}$

$\Rightarrow 2\log (V-1)+5\log (V+1)+7\log X=\log c$

$\Rightarrow \log (V-1{)}^2+\log (V+1{)}^5+\log X^7=\log c$

$\Rightarrow \log (V-1{)}^2(V+1{)}^5{X}^7=\log c$

$\Rightarrow (V-1{)}^2(V+1{)}^5{X}^7=c$

$\frac{(Y-X)(Y+X)}{X^7}{X}^7=c$

$\Rightarrow (y-x+1)(y+x-1{)}^5=c$