Given the fourth order homogeneous constant coefficient equation y\"\" + 10y\" +

Question

Given the fourth order homogeneous constant coefficient equation y"" + 10y" + 9y = 0 the auxiliary equation is ar^4 + br^3 + cr^2 + dr + e = 0. The roots of the auxiliary equation are (enter answers as a comma separated list). A fundamental set of solutions is (Enter the fundamental set as a commas separated list y_1, y_2, y_3, y_4). Therefore the general solution can be written as y = c_1y_1 + C_2y_2 + c_3y_3 + c_4y_4. Use this to solve the IVP with y(0) = 0, y'(0) = 1, y"(0) = 16, y'"(0) = -25 y(x) =

Explanation / Answer

1) The auxilliary equation is r4 + 0r3 + 10r2 + 0r + 9 =0

Hence the auxilliary equation is r4 + 10r2 + 9 =0

2) r4 + 9r2 + r2 + 9 = 0

=> r2 (r2 + 9) +1 (r2 + 9) =0

=> (r2 + 1)(r2 + 9) =0

=> r2 = -1, r2 = -9

=> r = +i, -i, +3i, -3i

3) Based on roots of auxilliary equation, the fundamental set of solutions is given by :

y1 = cosx, y2 = sinx, y3 = cos3x, y4 = sin3x

4) y= c1cosx + c2sinx+ c3cos3x + c4sin3x

y(0) = 0 => 0 = c1 + c3...a)

y' = -c1sinx + c2cosx -3c3sin3x +3c4cos3x

y'(0) = 1 => 1 = c2 + 3c4 ...b)

y'' = -c1cosx -c2sinx -3c3cos3x -3c4sin3x

y"(0) = 16 => 16 = c1 -3c3...c)

y"' = c1sinx -c2cosx +9c3sin3x -9c4cos3x

y"'(0) = -25 => -25 = -c2 - 9c4...(d)

a) -c) => -16 = 4c3 => c3 = -4

Hence c1 = 4 (Using a))

Also b) + d) => -24 = -6c4 => c4 = 4

Hence c2 = 1-12 = -11 (Using b))

So the solution to initial value problem is given by :

y(x) = 4cosx -11sinx -4cos3x +4sin3x

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