So I was blasting through my online homework and all of a sudden it changed the problems up on me without a good into to it. Here it is

Solve the following initial value problem:

y^(4)= - 3sin t +4cos t, with y'''(0)=9, y''(0)= - 4, y'(0)= -3, y(0)= 1

My line of thought had me turning it into

y(t)=3cost+4sint+C

and then

1=3cos(0)+4sin(0)+C
c=2

Then it asks for y=

At this point I enter 2 and it was wrong

Come on guys, help a feeble minded brother out

I assume by y^4 you mean y'''' (y quadruple prime or the 4th y derivative)?

What class are you taking?  It sounds like you're taking about a regular calculus class, but solving IVPs is stuff from a differential equations class.  If that's just thrown in without any explanation, then that's really dumb.

On the problem it looks like how I typed it so I am not really sure as I have never seen something written that way. The problems I was doing leading up to it were basically the intro to integration. It is only Calculus 1 so if the is from Differential Equations, this is incredibly stupid

What you're looking at is a 4th order, non-homogeneous differential equation initial value problem.

I noticed you said, "online homework." Someone got a little mixed up when posting your homework

I could help you solve it, but this is stuff you don't learn until near the end of a DE course. Don't bother with this one. This is not material from your class. If your professor doesn't already know, you should email them and tell them differential equations made their way to your homework.

This is doable with calc 1 material.
You have y''''.
Integrate y'''' to get y''', use initial value to solve for C.
Integrate y''' to get y'', use initial value to solve for C.
Integrate y'' to get y', use initial value to solve for C.
Integrate y' to get y, use initial value to solve for C.

Should get something like y = -3sin(t) + 4cos(t) +t^3 - 3, with no Diff Eq knowledge needed.

My answer (treating this as a DE) came out to be y = 1 - 4t + (3/2)t^2 -3sint + 4cost. wrong because it's been a while, and forgot to add particular solution on before messing with initial values

This particular problem CAN be done, as noted above, by Calc I methods. Only because the roots of the complementary are repeated 0's, however.

This question, however, was CLEARLY intended for a DE student as an exercise in both repeated roots and undetermined coefficients.

calc prob

Not sure if this will work.  It is a maple sheet I saved as html, I used the calc methods to work the problem.  I couldn't paste the text in here.

Yep, this.

Remember that each time you integrate, you're getting a different C, and that the C's that you already have need to be integrated as well.