(dy)/(dx) = (x-2) e^(-2y) and y(2)=ln2 what is y(x)?

Question

`(dy)/(dx) = (x-2) e^(-2y)` and
`y(2)=ln2` what is y(x)?

Answer 1

`(dy)/(dx) =(x-2)e^(-2y)`

`\int e^(2y) dy = \int (x-2)dx`

`(1)/(2) e^(2y)= (1)/(2) x^2-2x+c`
given y(2) = ln 2:
`(1)/(2) e^(2\ln2)= ((1)/(2) * 2^2)-(2*2)+c`

`2=2-4+c \therefore c=4`

`(1)/(2) e^(2y)= (1)/(2) x^2-2x+4`

`e^(2y)=x^2-4x+8`

`2y=\ln(x^2-4x+8)`

`y(x)= (1)/(2) \ln(x^2-4x+8)`