Question

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

Answer #1

For a linear oscillator (block on spring),
x(t) = xm cos (ωt + φ) (1)
v(t) = −ωxm sin (ωt + φ) (2)
a(t) = −ω
2xm cos (ωt + φ). (3)
(a) Draw and explain in words how you would use a circle diagram
to
connect x(t), v(t), and the phase (ωt + φ).
(b) What does ω represent? How is ω different for a linear
oscillator
than for a rolling or rotating object?

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...

In this problem, x = c1 cos t + c2 sin t is a two-parameter
family of solutions of the second-order DE x'' + x = 0. Find a
solution of the second-order IVP consisting of this differential
equation and the given initial conditions.
x(π/6) = 1 2 , x'(π/6) = 0
x=

4 Schr¨odinger Equation and Classical Wave Equation
Show that the wave function Ψ(x, t) = Ae^(i(kx−ωt)) satisfies
both the time-dependent Schr¨odinger equation and the classical
wave equation. One of these cases corresponds to massive particles,
such as an electron, and one corresponds to massless particles,
such as a photon. Which is which? How do you know?

Answer all parts of question 1.
1a.) Show that the solutions of x' = arc tan (x) + t cannot have
maxima
1b.) Show that the solution x(t) of the Cauchy problem x' = 2 +
sin(x), x(0) = 0, cannot vanish for t>0
1c.) Let Φ(t) be the solution of the ivp x' = tx - t^3, x(0) =
a^2 with a not equal to 0. Show that Φ has a minimum at t=0.

Solve the following differential equations
1. cos(t)y' - sin(t)y = t^2
2. y' - 2ty = t
Solve the ODE
3. ty' - y = t^3 e^(3t), for t > 0
Compare the number of solutions of the following three initial
value problems for the previous ODE
4. (i) y(1) = 1 (ii) y(0) = 1 (iii) y(0) = 0
Solve the IVP, and find the interval of validity of the
solution
5. y' + (cot x)y = 5e^(cos x),...

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