Solve the following differential equation for the particular

Solve the following differential equation for the particular solution. dy/dx=y(x^2+2) y(0)=2

dy/dx = y(x^2+2)

seperating the variables we get

dy/y = (x^2+2)dx

Integrating both sides we get

ln(y) = x^3/3 + 2x + C

Now using the initial condition that y(0) = 2, substituting the values we get

ln(2) = 0^3/3 + 2(0) + C

C = ln(2)

Hence the final answer is

ln(y) = 1/3 * x^3 + 2x + ln(2)