Exact solutions and the convergence of eulers method

[thedude36]

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=\phi(t)=(y₀-t₀)e^t-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. I've tried to look up the term exact solutions but haven't found anything of much use.

Homework 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).

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+ce^t​

however, i don't think this is the correct method. What should i be doing instead of this?

 

[LCKurtz]

Your method is fine. You just have a mistake calculating your last line. Re-check that -2-t.

 

[thedude36]

Your method is fine. You just have a mistake calculating your last line. Re-check that -2-t.

Thanks! I am not sure how i was integrating that wrong, but a quick check on a calculator quickly resolved the issue. I appreciate it!