im having trouble with this question - http://i.imgur.com/Ars4J1b.png - more specifically with part a, as i have a good idea how to go about b. given the initial value problem y' = 1-t+y , y(t0)=y0 show that the exact solution is y=[itex]\phi[/itex](t)=(y0-t0)et-t0+t we've only spoken of approximations in class and i've just been kind of guessing as to how i should go about it so far. Ive tried to look up the term exact solutions but havent found anything of much use. 2. Relevant equations integrating factor: if dy/dt+ay=g(t), then μ(t) is such that dμ(t)/dt=aμ(t). Multiply both sides of the equation dy/dt+ay=g(t) by μ(t) to obtain μ(t)dy/dt+ayμ(t)=μ(t)g(t). 3. The attempt at a solution rearranging the given formula, i was able to get dy/dt-y=1-t where a = -1. thus, dμ(t)/dt=-μ(t) making μ(t)=e-t. multiplying bothsides give e-tdy/dt-e-ty=e-t-te-t the left side can be obtained by the power-rule if the initial function was d(ye-t)/dt so we replace the leftside with this d(ye-t)/dt=e-t-te-t integrating bothsides and then solving for y gives y=-2-t+cet however, i dont think this is the correct method. What should i be doing instead of this?