Question

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.

Answer #1

All we need to do is find the auxiliary equation for the given differential equation and solve it like a normal quadratic equation. That is the auxiliary equation needs to be solved by either splitting the middle term or by the quadratic formula that is .

If
, then the case will be of **under damped
oscillation**. If
the case if of **over damped oscillation** and if
it
is **critically damped oscillation**.

1. The forced response for a first-order differential equation
with a constant forcing function is also referred to as the
steady-state solution.
True or false
2. If the roots of the characteristic equation for a
second-order circuit are real and equal, the network response
is:
a. overdamped
b. underdamped
c. critically damped
3. If the response of a second-order circuit is oscillatory, the
circuit is
a. underdamped
b. critically damped
c. overdamped
4. The forced response for a second-order circuit...

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.

Solve the special type second order differential
equation:
x2*d2y/dx^2+x(dy/dx)=1

Question 11:
What is the general solution of the following homogeneous
second-order differential equation?
d^2y/dx^2 + 10 dy/dx + 25.y =0
(a)
y = e 12.5.x (Ax + B)
(b)
y = e -5.x (Ax + B)
(c)
y = e -10.x (Ax + B)
(d)
y = e +5.x (Ax + B)
Question 12:
What is the general solution of the following homogeneous
second-order differential equation?
Non-integers are expressed to one decimal place.
d^2y/dx^2 − 38.y =0
(a)
y...

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)

solve differential equation ((x)2 - xy +(y)2)dx - xydy
= 0
solve differential equation (x^2-xy+y^2)dx - xydy =
0

Consider the nonlinear second-order differential
equation 4x"+4x'+2(k^2)(x^2)− 1/2 =0, where k > 0 is a constant.
Answer to the following questions.
(a) Show that there is no periodic solution in a simply connected
region R={(x,y) ∈ R2 | x <0}. (Hint: Use the corollary to
Theorem 11.5.1>>
If symply connected region R either contains no critical points of
plane autonomous system or contains a single saddle point, then
there are no periodic solutions. )
(b) Derive a plane autonomous system...

Use the Laplace transform to solve the given system of
differential equations. dx/dt=x-2y dy/dt=5x-y x(0) = -1, y(0) =
6

The indicated function y1(x) is a solution of the given
differential equation.
Use reduction of order or formula (5) in Section 4.2,
y2 = y1(x) e−∫P(x) dx y 2 1 (x) dx (5)
as instructed, to find a second solution y2(x). y'' + 100y = 0;
y1 = cos 10x
I've gotten to the point all the way to where y2 = u y1, but my
integral is wrong for some reason
This was my answer
y2= c1((sin(20x)+20x)cos10x)/40 + c2(cos(10x))

Consider the following system of differential equations dx/dt =
(x^2 + 2x + 1)(x^2 − 4x + 4) dy/dt = xy − 1
Which of the following is not an equilibrium point of the above
system? (A) (3, 1/3 ) (B) (−1, −1) (C) (1, 1) (D) (1, 3)

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