Question

The position x(t), of a damped oscillator with forcing satisfies the ordinary differential equation ,

i) where f(t) denotes the forcing on the oscillator. (i) If x(0) = 0, dx dt (0) = 1, f(t) = 4t and the Laplace transform of x(t) is denoted X(s) = L[x(t)], then show that

X(s) = 1 /(s + 2)^2 + 4 /s^2 (s + 2)^2

ii) Hence find x(t)

Answer #1

Consider the second order differential equation d2/dt^2 x + 6
dx/dt + 10x = 0. Classify the harmonic oscillator
(undamped, underdamped, critically damped, over damped). Justify
your answer.

Python: We want to find the position, as a function of time, of
a damped harmonic oscillator. The equation of motion is
m d2x/dt2 = -kx – b
dx/dt, x(0) = 0.5, dx/dt (0) = 0
Take m = 0.25 kg, k = 100 N/m, and b = 0.1 N.s/m.
Solve x(t) for t in the interval [0, 10T], where T = 2π/ω, and
ω2 = k/m.
please write the code.
Divide the interval into N = 104 intervals:...

Consider the driven damped harmonic oscillator
m(d^2x/dt^2)+b(dx/dt)+kx = F(t)
with driving force F(t) = FoSin(wt).
Consider the overdamped case
(b/2m)^2 < k/m
a. Find the steady state solution.
b. Find the solution with initial conditions x(0)=0,
x'(0)=0.
c. Use a plotting program to plot your solution for
m=1, k=0.1, b=1, Fo=0.25, and w=0.5.

Use the Laplace transform to solve the given system of
differential equations. dx dt = −x + y dy dt = 2x x(0) = 0, y(0) =
2

Consider the following initial value problem:
x′′−3x′−40x=sin(2t),x(0)=4,x′(0)=3
Using X for the Laplace transform of x(t), i.e., X=L{x(t)},,
find the equation you get by taking the Laplace transform of the
differential equation and solve for
X(s)=

Transform the differential equation x2d2y/
dx2 − xdy/dx − 3y = x 1−n ln(x), x > 0 to
a linear differential equation with constant coefficients. Hence,
find its complete solution using the D-operator method.

Find the particular solution of the differential equation that
satisfies the initial condition(s).
f "(x)=2, f '(2) = 5, f(2)=10

For the below ordinary differential equation with initial
conditions, state the order and determine if the equation is linear
or nonlinear. Then find the solution of the ordinary differential
equation, and apply the initial conditions. Verify your solution.
x^2/(y^2-1) dy/dx=(3x^3)/y, y(0)=2

Use Euler’s method to numerically solve the differential
equation dx/dt=0.3x−10 for 0≤t≤3 given that x=40 when t=0. Do not
do any rounding. Work must be shown

Find the Laplace transform of the given function:
f(t)=(t-3)u2(t)-(t-2)u3(t),
where uc(t) denotes the Heaviside function, which is 0 for
t<c and 1 for t≥c.
Enclose numerators and denominators in parentheses. For example,
(a−b)/(1+n).
L{f(t)}=
_________________ , s>0

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