Question

Q.3 (Applications of Linear Second Order ODE): Consider the ‘equation of motion’ given by ODE d2x + ω2x = F0 cos(γt)

dt2

where F0 and ω ̸= γ are constants. Without worrying about those
constants, answer the questions (a)–(b).

(a) Show that the general solution of the given ODE is [2 pts] x(t) := xc + xp = c1 cos(ωt) + c2 sin(ωt) + (F0 / ω2 − γ2) cos(γt).

(b) Find the values of c1 and c2 if the initial conditions are x(0) = 0 and x′(0) = 0, so that the general ?

solution in part (a) can be written explicitly as x(t) =( F0/ ω2 − γ2) (cos(γt) − cos(ωt) ).

Answer #1

A particle of mass m, is under the influence of a force F given
by
F = Fo [(sin ωt)ˆi + (cos ωt) ˆj]
where F0, ω are positive constants. If at t = 0 the particle is
at rest at the origin, find
(a) the equations of motion x (t) and y (t) of the particle,
and
(b) the work done by the force F from t = 0 to t = 2π/ω.

Show that
x(t) =c1cosωt+c2sinωt, (1)
x(t) =Asin (ωt+φ), (2) and x(t) =Bcos
(ωt+ψ) (3)
are all solutions of the differential equation d2x(t)dt2+ω2x(t)
= 0. Show that thethree solutions are identical. (Hint: Use the
trigonometric identities sin (α+β) =sinαcosβ+ cosαsinβand cos (α+β)
= cosαcosβ−sinαsinβto rewriteEqs. (2) and (3) in the form of Eq.
(1). To get full marks, you need to show the connection between the
three sets of parameters: (c1,c2), (A,φ), and (B,ψ).)
From Quantum chemistry By McQuarrie

For problems 3-6, note that if y(t) is a solution of the
homogeneous problem, then y(t−t0) is a solution as well, where t0
is a fixed constant. So, for example, the general solution of a
problem with complex roots can be expressed as y(x) = c1e µ(t−t0)
cos(ω(t − t0)) + c1e µ(t−t0) sin(ω(t − t0)) When initial conditions
are given at time t = t0 and not t = 0, expressing the general
solution in terms of t −...

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))

6) (8 pts, 4 pts each) State the order of each ODE, then
classify each of them as
linear/nonlinear, homogeneous/inhomogeneous, and
autonomous/nonautonomous.
A) Unforced Pendulum: θ′′ + γ θ′ + ω^2sin θ = 0
B) Simple RLC Circuit with a 9V Battery: Lq′′ + Rq′ +(1/c)q = 9
7) (8 pts) Find all critical points for the given DE, draw a phase
line for the system,
then state the stability of each critical point.
Logistic Equation: y′ = ry(1 −...

1) State the main difference between an ODE and a PDE?
2) Name two of the three archetypal PDEs?
3) Write the equation used to compute the Wronskian for two
differentiable
functions, y1 and y2.
4) What can you conclude about two differentiable functions, y1 and
y2, if their
Wronskian is nonzero?
5) (2 pts) If two functions, y1 and y2, solve a 2nd order DE, what
does the Principle of
Superposition guarantee?
6) (8 pts, 4 pts each) State...

1) The given family of functions is the general solution of the
differential equation on the indicated interval. Find a member of
the family that is a solution of the initial-value problem.
y = c1 + c2 cos(x) + c3 sin(x),
(−∞, ∞);
y''' + y' = 0, y(π) =
0, y'(π) = 8, y''(π)
= −1
y =
2) Two chemicals A and B are combined to form
a chemical C. The rate, or velocity, of the reaction is
proportional to the...

3. Consider the nonlinear oscillator equation for x(t) given by
?13 ?
x ̈+ε 3x ̇ −x ̇ +x=0, x(0)=0, x ̇(0)=2a
where a is a positive constant. If ε = 0 this is a simple
harmonic oscillator with frequency 1. With non-zero ε this
oscillator has a limit cycle, a sort of nonlinear center toward
which all trajectories evolve: if you start with a small amplitude,
it grows; if you start with a large amplitude, it decays.
For ε...

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