find general solution in semi homogenous method of dy/dx=x-y+1/x+y-1

Question
Differential equations
asked 2021-01-16
find general solution in semi homogenous method of \(\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={x}-{y}+\frac{{1}}{{x}}+{y}-{1}\)

Answers (1)

2021-01-17
The Given problem is \(\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=\frac{{{x}-{y}+{1}}}{{{x}+{y}-{1}}}\)
Let us substitute X=x and Y=y-1 then dX=dx and dY=dY
Hence the above Differential equations reduces to PSKdY/dX=(X-Y)/(X+Y) ->(X+Y)dY=(X-Y)dX ->XdY+YdY=XdX-YdY ->XdX+YdX+YdY-XdX=0 ->d(XY)+YdY-XdX=0 ->∫d(XY)+∫YdY-∫XdX=C ->XY+((Y^2/2))-((X^2)/2)=CZSK
Substituting the values of X and Y \(\displaystyle\to{x}{\left({y}-{1}\right)}+{\left(\frac{{1}}{{2}}\right)}{\left({\left({y}-{1}\right)}^{{2}}-{x}^{{2}}\right)}={c}\)
This is the required general solution.
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