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Diff Eq 3.3

 

<== Differential Equations
Number Question Answer
See the work
1. Find the general solution to the homogeneous differential equation
rac{d^2y}{dt^2} - 14rac{dy}{dt} + 65y = 0

The solution has the form
y = C_1 f_1(t) + C_2 f_2(t)
with f_1(t) = and f_2(t) =
Left to your own devices, you will probably write down the correct answers, but in case you want to quibble, enter your answers so that the functions are normalized with their values at t=0 equal to 0 and 1 (respectively), and they are expressed as simply as possible.

f_1(t) =e^(7t)(cos(4t))

f_2(t) = e^(7t)(sin(4t))

The work

Point Cost: 3
2.
Find the general solution to the homogeneous differential equation
rac{d^2y}{dt^2} - 18rac{dy}{dt} + 90y = 0

The solution has the form
y = C_1 f_1(t) + C_2 f_2(t)
with f_1(t) = and f_2(t) =
Left to your own devices, you will probably write down the correct answers, but in case you want to quibble, enter your answers so that the functions are normalized with their values at t=0 equal to 0 and 1(respectively), and they are expressed as simply as possible.
f_1(t) =e^(9t)cos(3t)
f_2(t) =e^(9t)sin(3t)
The work

Point Cost: 3
3. Find all values of k for which the function y=sin(kt) satisfies the differential equation y''+19y=0. Separate your answers by commas.
sqrt(19), -sqrt(19), 0 The work
4. Find y as a function of t if
36y'' + 108y' + 11y = 0 
y(0) = 6,    y'(0) = 8
y=
e^(-3/2t)((6cos(sqrt(4320)/72t))+18.62256696sin(sqrt(4320)/72t))
The work

Point Cost: 3
5. Find y as a function of t if
49y'' + 70y' + 25y = 0
y(0) = 5,     y' = 2.
y=
5e^(-70/98t)+39/7te^(-70/98t)
The work

Point Cost: 3
6. Find y as a function of t if
1600 y'' + 729y = 0,
y(0) = 7,     y'(0) = 5 

y=
7cos(27/40t)+200/27sin(27/40t)
The work

Point Cost: 3
7. Find y as a function of t if
16y'' = 0
y(0) = 5,    y'(0) = 7
y=
y=5+7t
The work

Point Cost: 3
8. Find y as a function of t if
y'' + 6y' + 58y = 0, y(0) = 4, y'(0) = 8.

y=
e^(-3t)(4cos(7t)+(20/7)sin(7t))
The work

Point Cost: 3
9. Find y as a function of t if
16y'' - 32y' + 16y = 0
y(1) = 4,   y'(1) = 2.
y=
2.207277e^t-0.735759te^t
The work

Point Cost: 3
10. Find the solution to initial value problem
rac{d^2y}{dt^2}  + 2rac{dy}{dt} + 1y  = 0, \ \ \     y(0) = 3, y'(0) = 2
The solution is .
3e^-t+5te^-t
The work

Point Cost: 3
11. Match the third order linear equations with their fundamental solution sets.

 1. ty'''-y'' = 0
 2. y'''+3y''+3y'+y = 0
 3. y'''-6 y'' +8 y' = 0
 4. y'''- y''- y'+ y = 0
 5. y'''+y' = 0
 6. y'''-2 y''+y'-2 y = 0


A. e^{2 t}, \ \cos(t), \ \sin(t)
B. e^t, \ te^t, \ e^{-t}
C. e^{-t}, \ te^{-t}, \ t^2e^{-t}
D. 1, \ e^{4 t}, \ e^{2 t}
E. 1, \ t, \ t^3
F. 1, \ \cos(t), \ \sin(t)

 1.
 2.
 3.
 4.
 5.
 6. 
The work
12. Find y as a function of x if
y''' - 20y'' + 24y' = 0,
y(0) = 1,    y'(0) = 3,    y''(0) = 5
 .
y(x)=
-1/24-7/12e^(6x)+13/8e^(4x)
The work

Point Cost: 3
13.

Find y as a function of x if

y''' + 4y' = 0,
y(0) = 1,    y'(0) = 0,    y''(0) = 16

y(x)=

5-4(cos(2x))
The work

Point Cost: 3
14.

Find y as a function of x if

y^{(4)} - 4 y''' + 4 y'' =  0,

 

y(0) = 6,        y'(0) = 7,         y''(0) = 4,         y'''(0) = 0.

y(x)=

3+3x+3e^(2x)-2xe^(2x)
The work

Point Cost: 3
15.

Find y as a function of x if

y''' - 3y'' - y' + 3y = 0,
y(0) = 4,       y'(0) = 8,        y''(0) = 20. 

y(x)=

2e^x+2e^(3x)
The work

Point Cost: 3