i have this homework that i failed at last week and i need to know the correct answers please

AE 280 Global Campus
Homework Three
Exercise 1 (a) Show that the given functions are solutions for the ODE
and
(b) form a basis on any interval. Use Wronskians.
ex , e?x , e2x ,
y 000 ? 2y 00 ? y 0 + 2y = 0
Exercise 2 Solve the given ODE. Show the details of your work.
y 000 + 25y 0 = 0
Exercise 3 Solve the given ODE with initial conditions. Show the details of your work.
4y 000 + 8y 00 + 41y 0 + 37y = 0, y(0) = 9, y(0)0 = ?6.5, y(0)00 = ?39.75
Exercise 4 Solve the given ODE with initial conditions. Show the details of your work.
x3 y 000 + xy 0 ? y = x2 , y(1) = 1, y(1)0 = 3, y(1)00 = 14
Exercise 5 Find the transform. Show the details of your work.
a)
f (t) = (a ? bt)2
b)
f (t) = cos2 (wt)
Exercise 6 Find the inverse Laplace transforms.
a)
5s + 1
F (s) = 2
s ? 25
b)
1
F (s) =
(s + a)(s + b)
Exercise 7 Find the inverse Laplace transforms.
a)
21
?
F (s) =
(s + 2)4
b)
6
F (s) =
(s + 1)3
c)
4
F (s) = 2
s ? 2s ? 3
1
Exercise 8 Solve the IVPs by the Laplace transfrom.
a)
y 00 ? y 0 ? 6y = 0, y(0) = 11,
y 0 (0) = 28
b)
y 00 ? 4y 0 + 4y = 0,
y(0) = 8.1,
Exercise 9 Find inverse Transforms by integration
a)
3
F (s) = 2
(s + s/4)
b)
F (s) =
1
s(s2 + ? 2 )
2
y 0 (0) = 3.9

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