Question

(3 points) Given the fourth order homogeneous constant coefficient equation y"" + 10y" + 9y-0 1) the auxiliary equation is ar br3+ cr2 + dr +e 2) The roots of the auxiliary equation are (enter answers as a comma separated list) 3) A fundamental set of solutions is Enter the fundamental set as a commas separated list yi , y2,y, y4). Therefore the general solution can be written as ycy c232 +c3ys c4y4- 4) Use this to solve the IVP with y(0) 0, y(0-3, y" (0)--16, y", (0--5 1 y(x)

Answer 1

Given differential equation is:

$y''''+10y''+9y=0$

1) The auxiliary equation for a fourth order homogeneous constant coefficient differential equation

$ay''''+by'''+cy''+dy'+ey=0$ is:

$ar^4+br^3+cr^2+dr+e=0$

Therefore the auxiliary equation is:

$r^4+10r^2+9=0$

2) the roots of this equation are:

$(r^2+1)(r^2+3)=0$

$\therefore r=\pm i, \pm 3i$

3) A fundamental set of solutions is:

$y_{1}=a\sin x, y_{2}=b\cos x, y_{3}=c\sin 3x, y_{4}=d\cos 3x$

Therefore the general solution is:

$y=a\sin x+b\cos x+c\sin 3x+d\cos 3x$

4) the first three derivatives of this are:

$y'=a\cos x-b\sin x+3c\cos 3x-3d\sin 3x$

$y''=-a\sin x-b\cos x-9c\sin 3x-9d\cos 3x$

$y'''=-a\cos x+b\sin x-27c\cos 3x+27d\sin 3x$

Substitute given initial conditions into these.

y(0)=0, y'(0)=3, y'''(0)=-16, y'''(0)=-51

y(0)=b+d=0

y''(0)=a+3c=3

y''(0)=-b-9d=-16

y'''(0)=-a-27c=-51

By solving these, we get

a=-3, b=-2, c=2, d=2

So, the solution to the initial value problem is:

$y(x)=-3\sin x-2\cos x+2\sin 3x+2\cos 3x$