Determine the integral equation equivalent to the initial value problem $$ y'(t)= t^2 + y(t)^4;\; y(0) = 1 $$ Could it be $y(t) = 1 + \int_0^t f(s,y(s))\mathrm ds?$ or $y(t) = 1 + \frac{t^3}{3} + \int_0^t f(s,y(s))\mathrm ds?$
As well as, the initial value problem $$ y''(t) + g(t,y) = 0,\, y(0)=y_0\;\;y'(0) = z_0 $$ which I know can be divided into two first order DE, however, my professor and book don't really explain on how to do this. Is there a standard way to formulate the integral equation from the initial value problem, if so what is it?