# Find fourth order differential equation

1. Oct 11, 2013

### kk1995

1. The problem statement, all variables and given/known data
Suppose that a fourth order differential equation has a solution y=-9e^(3x)xcos(x).

a. Find such differential equation, assuming that it is homogeneous and has constant coefficients.

b. Find the general solution to this differential equation. x is the independent variable.

3. The attempt at a solution
I tried to find the derivatives of y (y',y'',y''',y'''').
Then, I placed unknown constants (c1, c2, c3, c4, c5) on each order and made a matrix with each row having coefficients from values containing e^(3x)cosx, e^(3x)xcosx, e^(3x)sinx, and e^(3x)xsinx. I made the columns all have the same order (such as all values from y'' on one column). I also made c1=1 since otherwise the matrix becomes 4*5 and that cannot be solved for the unknown constants. However, this did not work out.

I would sincerely appreciate your help. I mean, I can do those functions that have y=e^rx and y=cos(x), but combination of these makes my head ache.

2. Oct 11, 2013

### SteamKing

Staff Emeritus
Have you tried writing cos (x) in terms of complex exponentials?

3. Oct 11, 2013

### LCKurtz

Think about what the solution whose DE had this characteristic equation$$(r-a)^2+b^2=0$$would be.

4. Oct 12, 2013

### HallsofIvy

Staff Emeritus
Great! Or course, in order to have $e^{3x}cos(x)$ and $e^{3x}sin(x)$ as solutions, the characteristic equation must have roots 3+ i and 3- i and so must have (x- 3- i) and (x- 3+ i) as factors. In order to have each of those times x as solutions, those must be double roots so the characteristic equation must be of the form $(x- 3- i)^2(x- 3+ i)^2= [(x- 3- i)(x- 3+ i)]^2=$$[(x- 3)^2+ 1]^2= (x^2- 6x+ 10)^2= x^4- 12x^3+ 56x^2- 12x+ 100$.

You can get the differential equation itself from that.

"columns"? "matrix"? Too advanced for me!

5. Oct 12, 2013

### kk1995

Thank you all for your replies. All of your replies really helped. The first two replies led me to Euler's formula, while the third one by HallsofIvy led the rest of the problem from the point of getting the values of r using Euler's formula. Note: the -12x should be -120x for the answer, just a point out at a typo. :)