Calculus: Early Transcendentals (7th Edition): Defining Second-Order Linear Differential Equations

Imagine you are an automotive engineer refining a luxury car's ride. As the vehicle glides over a bump, the interaction between the car's mass, the spring's stiffness, and the shock absorber's resistance is governed by a single mathematical structure: the Second-Order Linear Differential Equation. This is not just a formula; it is the language of vibration, stability, and control.

The Fundamental Structure

A second-order linear differential equation relates an unknown function $y(x)$ to its first and second derivatives. The term "linear" signifies that each instance of $y$, $y'$, and $y''$ appears only to the first power.

Standard Form

$$P(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x)y = G(x)$$

Where $P(x)$, $Q(x)$, $R(x)$, and $G(x)$ are continuous functions on a specific interval.

Classification of Equations

Homogeneous Equations: If $G(x) = 0$ for all $x$ in the interval, the equation is called homogeneous. These model systems in free vibration or equilibrium.
核心公式: $P(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x)y = 0$
Nonhomogeneous Equations: If $G(x) \neq 0$, the equation is nonhomogeneous. The function $G(x)$ represents an external forcing function (like hitting a pothole).

The Principle of Superposition

One of the most powerful tools in linear theory is the ability to construct complex solutions from simpler ones.

Theorem 3: Superposition

If $y_1(x)$ and $y_2(x)$ are both solutions of the linear homogeneous equation and $c_1, c_2$ are any constants, then the linear combination:

$y(x) = c_1y_1(x) + c_2y_2(x)$

is also a solution.

Finding the General Solution

To capture every possible solution of a homogeneous equation, we must ensure our two base solutions are linearly independent. This means neither is a constant multiple of the other (e.g., $e^x$ and $e^{2x}$ are independent, while $e^x$ and $2e^x$ are not).

Theorem 4: The General Solution

If $y_1$ and $y_2$ are linearly independent solutions on an interval and $P(x)$ is never 0, then the general solution is uniquely defined by:

$y(x) = c_1y_1(x) + c_2y_2(x)$

QUESTION 1

Which of the following defines a 'homogeneous' second-order linear differential equation?

The forcing function G(x) is equal to 0.

The coefficients P, Q, and R are all constant.

The solution y(x) is a linear function.

The second derivative y'' is equal to y.

QUESTION 2

If y₁(x) = sin(x) and y₂(x) = cos(x) are solutions to a homogeneous linear ODE, what is also a guaranteed solution?

y(x) = c₁sin(x) + c₂cos(x)

$y(x) = sin(x) * cos(x)$

$y(x) = sin(x) / cos(x)$

y(x) = sin²(x) + cos²(x)

QUESTION 3

What condition must be met for y₁(x) and y₂(x) to form a general solution?

They must be linearly independent.

They must be orthogonal.

They must both be exponential functions.

One must be the derivative of the other.

QUESTION 4

Are y₁(x) = eˣ and y₂(x) = 5eˣ linearly independent?

No, because y₂ is a constant multiple of y₁.

Yes, because they are distinct solutions.

No, because their derivatives are equal.

Yes, because the coefficient 5 is positive.

QUESTION 5

In the mechanical vibration model, what does the G(x) term represent?

External forcing (e.g., road bumps or motor input).

The mass of the system.

The stiffness of the springs.

Internal friction within the shock absorber.