Find y = f(x) by solving the differential equation dy/dx =2xy with the initial condition f(0) = 2

Answer 1

(1/y) dy = (2x) dx <--- separating the vars

ln y = x^2 + c <--- integrates both sides

y = e^(x^2 + c)

f(0)=2 ---> (0,2)

2 = e^(c)

ln 2 = c

y = e^(x^2) e^(ln 2)

= 2 e^(x^2)

CHECK:

y' = 2*e^(x^2) ( 2x)

= y*2x <--- y = 2e^(x^2)