I: Differential equations：練習問題

Determine the auxiliary equation and the nature of the roots for the second-order differential equation:
\(4\frac{d^2y}{dx^2} - 12\frac{dy}{dx} + 9y = 0\)

A particle's displacement \(x\) at time \(t\) satisfies the differential equation \(\ddot{x} + 6\dot{x} + 9x = 0\).
Given the initial conditions \(x(0) = 2\) and \(\dot{x}(0) = -1\), find the particular solution for \(x(t)\).

Consider the system of coupled first-order differential equations:
\(\frac{dx}{dt} = 4x - 2y\)
\(\frac{dy}{dt} = x + y\)
Which of the following represents the correct second-order differential equation for \(x\) in terms of \(t\)?

Find the integrating factor, \(I(x)\), for the first-order linear differential equation:
\(\frac{dy}{dx} + \frac{2x}{1+x^2}y = e^x\)

Find the general solution of the differential equation:
\(\frac{d^2y}{dx^2} + 4y = 8x^2\)

Find the general solution of the first-order differential equation:
\(\frac{dy}{dx} - 4y = e^{3x}\)

A particle moves such that its displacement \(x\) satisfies the second-order differential equation:
\(\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 4x = 0\)
Describe the type of damping exhibited by this system and state the form of the general solution for \(x(t)\).

Find the particular solution of the differential equation:
\(\frac{d^2y}{dx^2} + y = 2e^x\)
given the initial conditions that when \(x=0\), \(y=2\) and \(\frac{dy}{dx} = 1\).

A particle moves in simple harmonic motion such that its displacement \(x\) satisfies the equation \(\frac{d^2x}{dt^2} + 25x = 0\).
At \(t=0\), the particle is at the origin and is moving with a velocity of \(10 \text{ m s}^{-1}\).
Find the expression for \(x\) in terms of \(t\) and state the amplitude of the motion.

A particle of mass 0.5 kg is attached to a light spring and moves in a medium where the resistance is proportional to its velocity. The displacement \(x\) of the particle at time \(t\) satisfies the differential equation:
\(\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 25x = 0\)

(a) Show that the motion is underdamped.
(b) Given that at \(t = 0\), \(x = 0.2\) and \(\frac{dx}{dt} = 0\), find the particular solution for \(x\) in terms of \(t\).
(c) State the time at which the particle first returns to the equilibrium position.