[tex] \frac{dy}{dx} =(x-2)e^{-2y}[/tex]
[tex]\int e^{2y} dy = \int (x-2)dx[/tex]
[tex] \frac{1}{2} e^{2y}= \frac{1}{2} x^2-2x+c[/tex]
given y(2) = ln 2: [tex] \frac{1}{2} e^{2\ln2}= (\frac{1}{2} \times 2^2)-(2\times2)+c[/tex]
[tex]2=2-4+c \therefore c=4[/tex]
[tex] \frac{1}{2} e^{2y}= \frac{1}{2} x^2-2x+4[/tex]
[tex]e^{2y}=x^2-4x+8[/tex]
[tex]2y=\ln(x^2-4x+8)[/tex]
[tex]y(x)= \frac{1}{2} \ln(x^2-4x+8)[/tex]
