Question

Consider the following.

x'' + 4x' + 13x = 0, x(0) = 1, x'(0) = 1

Write the solution in the form

x(t) = Re^{αt} sin(ωt + ϕ).

x(t) = _____________

Graph it on the interval

0 ≤ t ≤ 10.

Answer #1

The above answer can be further modified or rearranged to the required form.

consider the following mode for simple harmonic motion in a mass
spring sysyem
4x"(t)+12x(t)=0
x(0)=1
x'(0)=1
compute solution x(t)
amplitude and period of motion

Differential Equations
(a) Write the function
x(t)=−2cos(4t)+5sin(4t)x(t)=−2cos(4t)+5sin(4t) in the form
x(t)=Acos(ωt−ϕ)x(t)=Acos(ωt−ϕ).
x(t)=
(b) Write the function x(t)=2cos(4t)+5sin(4t)x(t)=2cos(4t)+5sin(4t)
in the form x(t)=Acos(ωt−ϕ)x(t)=Acos(ωt−ϕ).
x(t)=
(c) Write
x(t)=e−3t(−2cos(4t)+5sin(4t))x(t)=e−3t(−2cos(4t)+5sin(4t)) in the
form y(t)=Aeαtcos(ωt−ϕ)y(t)=Aeαtcos(ωt−ϕ).
x(t)=

Two waves, y1(x,t) and
y2(x,t), travel on the same piece of
rope and combine to produce a resultant wave of the form
y(x,t) = 8.000 sin(4.000x +
1.000t + 0)cos(1.000x + 3.000t + 0). The
first wave is y1(x, t ) = 4.000
sin(3.000x + (-2.000)t), while the second wave
has the form y2(x, t ) = A
sin(kx ± ωt+ϕ), where x is
measured in m and t in seconds. Determine the values of
the constants in the second...

use Laplace transformations to solve initial value problem
x''+4x=cos(t), x(0)=0, x'(0)=0
set up the appropriate form of a particular solution, do not
determine coefficients,
y''+6y'+13y=e-3x cos2x
find the Laplace transformation of the following function
f(t)=4cos(2t) + 7t3 - 5e-3t

Verify the quotient rule for each of the following:
1. f(x)=4x^5-4x^4+13x^3-2x^2+7x+6/x^2-x+2
2. f(x)=20x^8-28x^6+29x^4-35x^2+5/5x^4-7x^2+1

Consider the function f(x)=6x^2−4x+11, on the interval
, 0≤x≤10.
The absolute maximum of f(x) on the given interval is at x =
The absolute minimum of f(x) on the given interval is at x =

f(x) = 2x^2 -4x-1
write in certex form and sketch graph
solve for x if [2x-1] <= 3

(* Problem 3 *)
(* Consider differential equations of the form a(x) + b(x)dy
/dx=0 *) \
(* Use mathematica to determin if they are in Exact form or not.
If they are, use CountourPlot to graph the different solution
curves 3.a 3x^2+y + (x+3y^2)dy /dx=0 3.b cos(x) + sin(x) dy /dx=0
3.c y e^xy+ x e^xydy/dx=0

Solve the following
a) 2 cos^2(4x) + 5 cos(4x) + 2 = 0.
b) arctan(3x + 3) = π/4
c) 2^1+sin^2(x) = 4^sin(x)
d) ln(x + 3) = ln(x) + ln(3)

Solve the initial value problem
x′=−3x−y,
y′= 13x+y,
x(0) = 0,
y(0) = 1.

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