Question

Consider the initial value problem, ay''+by'+cy=0, y(0)=d, y'(0)=f where a,b,c,d and f are constants which one of the following could be a solution to the initial value problem? Give breif exlpanation to why the correct answer can be a solution, and why the others can not possibly satisfy the equation.

a. sin(t)+e^t

b. cost+e^tsint

c. cost+1

d. e^tcost

Answer #1

3. Consider a second order linear homogeneous equation Ay'' +
By' + Cy = 0 Suppose that e^at, e^bt and e^ct are solutions (where
a, b, c are constants). A. Show that e^at + e^bt + e^ct is also a
solution. B. Show that two of the numbers among a, b, c are
equal.

a) The homogeneous and particular solutions of
the differential equation ay'' + by' + cy = f(x) are, respectively,
C1exp(x)+C2exp(-x) and 3x^3. Give the complete solution y(x) of the
differential equation.
b) If the force f(x) in the equation given in a)
is instead f(x) = f1(x) + f2(x) + f3(x), where f1(x), f2(x), and
f3(x) are generic forces, what would be the particular
solution?
c) The homogeneous solution of a forced oscillator
is cos(t) + sin(t), what is the...

Consider the initial value problem my′′+cy′+ky=F(t), y(0)=0,
y′(0)=0 modeling the motion of a spring-mass-dashpot system
initially at rest and subjected to an applied force F(t), where the
unit of force is the Newton (N). Assume that m=2 kilograms, c=8
kilograms per second, k=80 Newtons per meter, and F(t)=60cos(8t)
Newtons. Solve the initial value problem. y(t)= help (formulas)
Determine the long-term behavior of the system. Is limt→∞y(t)=0? If
it is, enter zero. If not, enter a function that approximates y(t)
for...

Consider the following initial value problem.
y′ + 5y =
{
0
t ≤ 2
10
2 ≤ t < 7
0
7 ≤ t < ∞
y(0) = 5
(a)
Find the Laplace transform of the right hand side of the above
differential equation.
(b)
Let y(t) denote the solution to the above
differential equation, and let Y((s) denote the
Laplace transform of y(t). Find
Y(s).
(c)
By taking the inverse Laplace transform of your answer to (b),
the...

Choose C so that y(t) = −1/(t + C) is a solution to the initial
value problem
y' = y2 y(2) = 3.
Verify that the given formula is a solution to the initial value
problem.
x′ = −y, y′ = x, x(0) = 1, y(0) = 0: x(t) = cost, y(t) = sin
t

Find the solution of the given initial value problem: y " + y =
f(t); y(0) = 6, y' (0) = 3 where f(t) = 1, 0 ≤ t < π 2 0, π 2 ≤
t < ∞

Let y = y ( t
) be the solution to the initial value problem
t
d y d t + 2 y = sin t , y ( π ) = 0
Find the value
of

Consider the BVP y′′=-cy, y(0)=0 where c>0, y(b)=0.
For each c, find b>0 such that the BVP has at least two
different solutions y1(t) and y2(t). Exhibit the solutions.

Consider the undamped spring equation
y'' + cy = sin(2t).
(a) For what value of c does resonance occur? Compute the
solution at resonance with y(0) = 1 and y' (0) = 0.
(b) For what values of c is there a beat with frequency 0.1 Hz?
(The beat frequency is defined as (|µ-w|)/2 where µ is the natural
frequency of the spring and ! is the forcing frequency.)

Consider the undamped spring equation
y'' + cy = sin(2t).
(a) For what value of c does resonance occur? Compute the
solution at resonance with y(0) = 1 and y' (0) = 0.
(b) For what values of c is there a beat with frequency 0.1 Hz?
(The beat frequency is defined as (|µ-w|)/2 where µ is the natural
frequency of the spring and ! is the forcing frequency.)

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