1) Solve y' + (1/t) y = t^3. Integrating factor =exp ∫(1/t)dt =exp (ln|t| + k) =exp (ln|t|) (take constant of integration k=0) =|t| ... and then I've found that the gerenal solution is: y = 1/|t| + [c + ∫(from 0 to t) |s| s^3 ds] Is this the correct final answer and is there any way to simply this? Can I get rid of all the absolute values? 2) Solve the initial value problem ty'+2y=4t^2, y(1)=2. Integrating factor=t^2 ... General solution is y = t^2 + c/t^2 Put y(1)=2 => c=1 So the solution to the initial value problem is y=t^2 + 1/t^2, t>0. Note that the function y=t^2 + 1/t^2, t<0 is NOT part of the solution of this initial value problem. ============================== I have no idea (red part) why you have to put the restriction t>0, and why is the part for t<0 definitely NOT part of the solution? What's the problem here? This example is driving me crazy... I am a beginner of this subject, and I hope that someone would be nice enough to explain these. Thanks!