Differential Equations：練習問題

Consider the second-order homogeneous differential equation \(\frac{d^2y}{dx^2} + 9y = 0\). Which of the following expressions represents the general solution for \(y\) in terms of the arbitrary constants \(A\) and \(B\)?

A mass-spring-damper system is modelled by the differential equation \(\frac{d^2x}{dt^2} + k \frac{dx}{dt} + 16x = 0\), where \(k\) is a positive damping constant. For which value of \(k\) does the system exhibit critical damping?

A non-homogeneous second-order differential equation is given by \(\frac{d^2y}{dx^2} - 4 \frac{dy}{dx} + 4y = 5e^{2x}\). What is the correct form for the particular integral, \(y_p\), given that the auxiliary equation has a repeated root?

Find the integrating factor, \(I(x)\), required to solve the first-order linear differential equation \(x \frac{dy}{dx} - 4y = x^5\) for \(x > 0\).

A system of coupled first-order linear differential equations is defined by:
\(\frac{dx}{dt} = 2x + y\)
\(\frac{dy}{dt} = x - 2y\)
By eliminating the variable \(y\), determine the auxiliary equation used to find the general solution for \(x(t)\).

Find the integrating factor for the first-order linear differential equation \( \frac{dy}{dx} + \frac{3}{x}y = x^2 \), where \( x > 0 \).

The general solution of a second-order homogeneous differential equation is \( y = (A + Bx)e^{-2x} \). State the auxiliary equation and identify the value of its discriminant.

Determine the particular integral for the differential equation \( \frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = 12e^{2x} \), given that the roots of the auxiliary equation are \( 2 \) and \( 3 \).

A first-order differential equation is given by \( \frac{dy}{dx} + \frac{2}{x}y = 3x^2 \) for \( x > 0 \).
(a) Find the general solution for \( y \) in terms of \( x \).
(b) Given that \( y = 1 \) when \( x = 1 \), find the particular solution, expressing \( y \) explicitly as a function of \( x \).

The motion of a damped system is modeled by the differential equation:
\( \frac{d^2y}{dt^2} + 4\frac{dy}{dt} + 4y = 8t^2 \)
(a) Find the complementary function for this differential equation.
(b) Find a particular integral for the differential equation.
(c) State the general solution for \( y \).
(d) Find the particular solution for the system given the initial conditions \( y(0) = 1 \) and \( \dot{y}(0) = 0 \).
(e) Describe the nature of the damping in this system.