Find the particular solution of the differential equation

Question

particular solution of the differential equation dy/dx=(x−7)e^(−2y).

Answer 1

y' = (x-7)e^(-2y)
e^2y y' = x-7
e^2y dy = x-7 dx
1/2 e^2y = 1/2 x^2 - 7x + c
e^2y = x^2 - 14x + c
2y = log(x^2-14x+c)
y = 1/2 log(x^2-14x+c)

Pick a c that fits a particular solution