eqnarray*}
dydt = f(t,y) y(t_0) = _0
y(t) -y(0) = _0^t f(s,y(s)) ds
Provided f is bounded at the end of this time interval, one can continue the process for another short time. The basic existence statement is:
Theorem If |f(t,y)-f(t,z)| ;SPMlt; L |y-z|, then there exists a unique solution to the IVP.
Examples:
y' = -y y(0)=1
y' = y^2 y(0)=1
y'' + k^2 y = 0 y(0)=1, y'(0)=0
y_1=y, y_1'=y_2 => Y=(y_1, y_2)^t Y'=AY