if g'(x)=2g(x) and g(-1)=1, then g(x)=
By definition, g'(x) = dg/dx.
Then, dg/dx = 2g
Or, dg/g = 2dx
Integrating,
int dg/g = int 2dx
ln(g) = 2x + ln(k) (where ln(k) is the constant of integration)
g(x) = k.e^(2x)
Since g(-1) = 1, then
1 = k.e^(-2)
k = e^2
So, g(x) = e^(2x+2)