xdy/dx=y+xy
x dy/dx = y(x+1)
x dy = y(x+1) dx
separate the variables
dy/y = (x+1)/x dx
dy/y = (1+1/x) dx
Integrate both sides
ln y = x + ln x +c1
y = e^(x+ln x+c1)
y = e^x e^ln x e^c1
let e^c1 = C
y = C x e^x
Divide both sides by xy:
1/y * dy/dx = 1/x + 1
1/y * dy = (1/x + 1)dx
Integrate both sides:
ln|y| = ln|x| + x + C
Now make y the subject, raise both sides to the base of e:
|y| = e^[ln|x| + x + C]
Simplifying:
|y| = e^ln|x| * e^(x) * e^C
As C is just a constant, e^C is also a constant so we can replace e^C with C:
y = Ce^(x)
I am not sure if you worded the question correctly but if this is what you meant I'll answer this:
F(x) = y+xy now dy/dx of F(x) = x+1
x(dy/dx) = y(1 + x)
dy/y = (1 + x)/x dx
Integrating both sides:
ln|y| = ln|x| + x + C
y = C*x*e^(x)
Please Help!
