Elementary Differential Equations and Boundary Value Problems: From Higher-Order Equations to First-Order Systems

The transition from higher-order differential equations to first-order systems represents a profound shift in perspective. Instead of tracking a single variable’s acceleration, we evolve a state-space vector representing position, velocity, and higher derivatives simultaneously. Any $n$-th order linear equation can be decomposed into a coupled system of $n$ first-order equations, allowing us to deploy the full power of matrix algebra.

1. The Reduction of Order Method

To transform the $n$-th order scalar equation $y^{(n)} = F(t, y, y', \dots, y^{(n-1)})$, we define a set of auxiliary variables:

$$x_1 = y, x_2 = y', \dots, x_n = y^{(n-1)}$$

This substitution leads to the vector equation $\mathbf{x}' = \mathbf{f}(t, \mathbf{x})$. For a classical mechanical oscillator described by $$mu'' + \gamma u' + ku = F(t)$$, the transformation results in:

$x_1' = x_2$
$x_2' = -\frac{k}{m}x_1 - \frac{\gamma}{m}x_2 + \frac{1}{m}F(t)$

Example 1: Spring-Mass Transformation

Problem

The motion of a certain spring-mass system is described by the second-order differential equation $u'' + \frac{1}{8}u' + u = 0$. Rewrite this equation as a system of first-order equations.

Substitution

Let $x_1 = u$ (position) and $x_2 = u'$ (velocity). Thus, $x_1' = x_2$.

Matrix Form

Substituting into the ODE: $x_2' + \frac{1}{8}x_2 + x_1 = 0 \Rightarrow x_2' = -x_1 - \frac{1}{8}x_2$.

$$\mathbf{x}' = \begin{pmatrix} 0 & 1 \\ -1 & -1/8 \end{pmatrix} \mathbf{x}$$

2. Coupled Physical Systems

While the reduction of order is a mathematical convenience for single equations, systems of equations arise naturally in complex environments:

Mechanical Systems: Multi-mass systems (like Figure 7.1.1) involve coupled forces where the movement of one mass affects the other via Hooke's Law.
Interconnected Tanks: Fluid flow between tanks (Figure 7.1.6) relies on the Conservation of Mass, where the rate of change of salt in Tank 1 depends on the concentration in Tank 2.
Electrical Circuits: Using constitutive relations $$V = RI, C \frac{dV}{dt} = I, L \frac{dI}{dt} = V$$, we build systems that describe the simultaneous evolution of voltage and current across inductors (L), capacitors (C), and resistors (R).

🎯 Core Principle

By treating derivatives as independent variables in a vector, we transform the complexity of "rate of change of rate of change" into a geometric rotation and scaling in state-space.

QUESTION 1

Transform the equation $u'' + 0.5u' + 2u = 0$ into a first-order system. What is the correct system of equations?

$x_1' = x_2, x_2' = -2x_1 - 0.5x_2$

$x_1' = x_2, x_2' = 2x_1 + 0.5x_2$

$x_1' = -0.5x_1, x_2' = -2x_2$

$x_1' = x_2, x_2' = -0.5x_1 - 2x_2$

QUESTION 2

For the IVP $u'' + 0.25u' + 4u = 2\cos(3t)$ with $u(0) = 1, u'(0) = -2$, what are the transformed initial conditions for the vector $\mathbf{x}(0)$?

$x_1(0) = 1, x_2(0) = -2$

$x_1(0) = -2, x_2(0) = 1$

$x_1(0) = 1, x_2(0) = 0$

$x_1(0) = 4, x_2(0) = 0.25$

QUESTION 3

In the interconnected tank system (Figure 7.1.6), if Tank 1 drains into Tank 2, why does the system of equations become 'coupled'?

Because the rate of change in Tank 2 depends on the salt concentration of Tank 1.

Because the gravity constant is the same for both tanks.

Because the total volume of water must remain exactly constant.

Because only one tank has an external salt source.

QUESTION 4

Which component in an RLC circuit is typically associated with the first-order relation $L \frac{dI}{dt} = V$?

The Inductor

The Capacitor

The Resistor

The Voltage Source

QUESTION 5

If a matrix $\mathbf{A}$ representing a system has a fundamental matrix $\Psi(t)$, how is the special fundamental matrix $\Phi(t)$ with $\Phi(0) = \mathbf{I}$ calculated?

$\Phi(t) = \Psi(t)\Psi(0)^{-1}$

$\Phi(t) = \Psi(t) + \mathbf{I}$

$\Phi(t) = \int \Psi(t) dt$

$\Phi(t) = \Psi(0)\Psi(t)^{-1}$