1. The problem statement, all variables and given/known data This was an example the teacher gave us to do in class. Unfortunately, I was a little behind on the notes so was not able to copy down the steps to solving this one. I did, however, get an answer problem: y'' + y = e^x + x^3, initial conditions y(0)=2, y'(0)=0 2. Relevant equations 3. The attempt at a solution The answer my prof gave was y=(3/2)cos(x)+(11/2)sin(x)+(1/2)e^(x)+x^3-6x I get how he got the complimentary part, but the particular solution I feel I am very close too, and may have just made some slightly wrong assumption as a 'trial' solution. what I did for a trial to the particular solution was... yp=Ae^x +Bx^3+Cx^2+Dx+E and set this equal too e^x + x^3. I found the second derivative to be (Ae^x + 6Bx + 2c). So far I have: (Ae^x + 6Bx + 2c) + (Ae^x +Bx^3+Cx^2+Dx+E) = e^x + x^3 I then gathered terms and solved for coefficients to get A=.5, B=1, C=0, D=-6, and E=0. simplifying I ended up with e^x +x^3 = e^x +x^3.... so I did something wrong somewhere I guess. I understand everything except how my prof got (1/2)e^(x)+x^3-6x as the particular solution. I even got all the terms he listed as the particular solution, but some of them ended up canceling out so my particular I believe is wrong.