Second order differential eq problem (from a calc2 class)

[icosane]

Homework Statement

y''+16y=9xe^(4x)

y(0)=0
y'(0)=0

find the solution, y(x) to the differential equation

The Attempt at a Solution

I found the particular solution to the right side of the equation, which is correct,

yp= .28125x-.0703125e^(4x)

For the left hand side of the equation I ended up with +- 4i, so using 4 as the beta value plugged it into,

y=Acos(4x)+Bsin(4x)

But plugging back into y''+16y I found it was the complementary equation... but does it even matter because there are no sines or cosines on the right hand side of the equation? I tried writing out the solution as

y = Acos(4x)+Bsin(4x) + .28125x-.0703125e^(4x)

Then solving it like an initial value problem but the computer won't take my answer. Any help would be greatly appreciated.

 

[Cyosis]

It is an initial value problem. Show us your steps,also show us how you got yp. Why is it in decimal form?

 

[HallsofIvy]

Homework Statement

y''+16y=9xe^(4x)

y(0)=0
y'(0)=0

find the solution, y(x) to the differential equation

The Attempt at a Solution

I found the particular solution to the right side of the equation, which is correct,

yp= .28125x-.0703125e^(4x)

This does NOT satisfy the differential equation! Without even doing any calculations I can see that the second derivative is a constant times e^(4x) so even after adding 16y you can NOT get "xe^(4x)". Try a particular solution of the form (Cx+ D)e^(4x).

For the left hand side of the equation I ended up with +- 4i, so using 4 as the beta value plugged it into,

y=Acos(4x)+Bsin(4x)

But plugging back into y''+16y I found it was the complementary equation... but does it even matter because there are no sines or cosines on the right hand side of the equation? I tried writing out the solution as

y = Acos(4x)+Bsin(4x) + .28125x-.0703125e^(4x)

Then solving it like an initial value problem but the computer won't take my answer. Any help would be greatly appreciated.

Your general solution for the equation is correct. The particular solution is not. Try again with my suggestion. If you cannot get the correct answer, come back and please show your work.

 

[icosane]

This is homework for my calc 2 class, which is evaluated by typing in my answers on a website. Once I type in an answer and get it correct it clears my answer and gives the "correct" answer in decimal form. So that is why it is in decimal form.

This does NOT satisfy the differential equation! Without even doing any calculations I can see that the second derivative is a constant times e^(4x) so even after adding 16y you can NOT get "xe^(4x)". Try a particular solution of the form (Cx+ D)e^(4x).

After figuring out the particular solution and getting it correct I then recopied it off the computer screen on the next page, but copied it wrong.

yp(x)= (0.28125*x+(-0.0703125))*exp(4*x)

Is the correct particular solution. I got it right, then re-copied it in decimal form wrong . Turns out that's all I did wrong and I got the correct answer. Thanks for pointing that out, Ivy.