The general first order ivp (initial value problem) is dy/dt = f(t,y) y(t 0 ) = y 0 (1) Where f(t,y) is a known function and the values in the initial condition are also known numbers. The second theorem in the intervals of validity we know that if f and fy are continuous functions then there is a unique solution to the ivp in some interval surrounding t = t 0 . Our goal is to approximate the solution neat t - t 0 . We know that the value of the solution at t = t 0 from the initial condition. We also know the value of the derivative at t = t 0 . We get this by plugging the initial condition into f(t,y) into the differential equation itself. Next we can write the equation of the tangent line to the solution at t = t0.
