Second-order Linear ODE's with Constant Coefficients II

If the only tool you have is a hammer, you tend to see every problem as a nail, Abraham Maslow.

Recall

Definition. A differential equation is an equation that involves one or more dependent variables, their derivatives with respect to one or more independent variables, and the independent variables themselves, e.g., $\frac{dy}{dx} = 3x +5y, y’ + y = 4xcos(2x), \frac{dy}{dx} = x^2y+y, etc.$

It involves (e.g., $\frac{dy}{dx} = 3x +5y$):

• Dependent variables: Variables that depend on one or more other variables (y).
• Independent variables: Variables upon which the dependent variables depend (x).
• Derivatives: Rates at which the dependent variables change with respect to the independent variables, $\frac{dy}{dx}$

First-Order Linear Ordinary Differential Equations (ODEs)

Definition. A first-order linear ordinary differential equation is an ordinary differential equation (ODE) involving an unknown function y(x), its first derivative y′, and functions of the independent variable x, which can be written in the general form:: a(x)y' + b(x)y = c(x) where:

• y = y(x) is the unknown function of the independent variable x.
• y′ = $\frac{dy}{dx}$ is the derivative of y with respect to x.
• a(x), b(x), and c(x) are known functions of x.

These equations are termed “linear” because the unknown function y and its derivative y’ appear to the first power and are not multiplied together or composed in any nonlinear way.

If the function c(x)=0 for all x in the interval of interest, the equation simplifies to: a(x)y’ + b(x)y = 0. Such an equation is called a homogeneous linear differential equation.

The Existence and Uniqueness Theorem provides crucial insight into the behavior of solutions to first-order differential equations ODEs. It states that if:

• The function f(x, y) (the right-hand side of the ODE) in the equation y’ = f(x, y) is continuous in some neighborhood around a point (x₀, y₀) and
• Its partial derivative with respect to y, denoted as $\frac{∂f}{∂y}$, is also continuous near (x₀, y₀).

Then, the differential equation y' = f(x, y) has a unique solution to the initial value problem through the point (x₀, y₀), meaning that it satisfies the initial condition y(x₀) = y₀.

This theorem ensures that under these conditions, the solution exists and is unique near x = x₀.

Second-order Linear Homogeneous ODE’s with Constant Coefficients

A second-order linear homogeneous differential equation ODE with constant coefficients is a differential equation of the form: y'' + Ay' + By = 0, where A and B are constants.

To solve this ODE, we seek two linearly independent solutions y₁(t) and y₂(t). The general solution is then a linear combination of these solutions: $c_1y_1 + c_2y_2$ where c₁ and c₂ are two arbitrary constants determined by initial conditions. The key to solving the ODE is the characteristic equation, whose roots determine the behavior of the solutions.

Complex Characteristic Roots

In the study of second-order linear homogeneous ordinary differential equations (ODEs) with constant coefficients, the nature of the roots of the characteristic equation plays a crucial role in determining the form of the general solution. When the characteristic equation has complex roots, the solutions involve exponential functions combined with sine and cosine functions.

Consider the second-order linear ODE: y’’ + Ay’ + By = 0. To solve this ODE, we assume a solution of the form: $y = e^{rt}$ where r is a constant to be determined. Substituting or plugging $y = e^{rt}$ into the differential equation, we obtain:

$r^2e^{rt} + Are^{rt} + Be^{rt} = 0$ ⇒[Cancelling the common factor $e^{rt}$ (which is never zero), we obtain the characteristic equation associated with the differential equation:] $r^2 + Ar + B = 0$.

If the characteristic equation has complex roots, say r = a ± bi, the general solution involves sine and cosine terms.

We get two complex solutions, one of them is $y=e^{(a+bi)t} = e^{at}e^{ibt} =[\text{Using Euler’s formula}] e^{at}(cos(bt) + isin(bt))$⇒ Its real part is $e^{at}cos(bt)$ and its imaginary part is $e^{at}sin(bt)$ ⇒[Theorem. The real and imaginary parts of a complex solution to a linear ODE with real coefficients are themselves real solutions.] $y_1 = e^{at}cos(bt)$ and $y_2 = e^{at}sin(bt)$ are two independent real solutions ⇒ Thus, the general solution is: y = $c_1y_1 + c_2y_2 = e^{at}(c_1cos(bt)+c_2sin(bt)) =$[Using the trigonometry identity a·cos(θ)+b·sin(θ) = c·cos(θ - Φ)] $e^{at}(c·cos(bt- Φ))$ where:

• c = $\sqrt{c_1^2+c_2^2}$
• $Φ = arctan(\frac{c_2}{c_1})$

Suppose we have chosen $y=e^{(a-bi)t}=e^{at}(cos(-bt)-isin(-bt))$⇒ Its real part is $e^{at}cos(-bt) = e^{at}cos(bt)$ and its imaginary part is $e^{at}sin(-bt) = -e^{at}sin(bt)$ ⇒[Theorem. The real and imaginary parts of a complex solution to a linear ODE with real coefficients are themselves real solutions.] $y_1 = e^{at}cos(bt)$ and $-y_2 = e^{at}sin(bt)$ are two independent real solutions⇒ Thus, the general solution is: y = $c_1y_1 + c_2y_2$. Although the sign of y₂ change, this does not affect the general solution because the constants c₁ and c₂ absorb the difference.

Alternative Approach Using Complex Constants

Alternatively, we can write the solution directly using complex constants. The general solution is: $y = C_1e^{(a+bi)t}+C_2e^{(a-bi)t}$ where C₁ and C₂ are complex constants. To find real solutions, we need the imaginary parts of this expression to cancel out.

Let’s express C₁ and C₂ as complex numbers: C₁ = c₁ + id₁ and C₂ = c₂ + id₂ where c₁, c₂, d₁, d₂ are real numbers, and i is the imaginary unit.

Now, substitute these into the solution:

$y = (c_1+id_1)e^{(a+bi)t}+(c_2+id_2)e^{(a-bi)t}$ ⇒ [Factoring out e^at] $y = e^{at}[(c_1+id_1)e^{bit}+(c_2+id_2)e^{-bit}]$ [Using Euler’s formula. e^bit= cos(bt) + isin(bt), e^-bit= cos(bt) - isin(bt)] $y = e^{at}[(c_1+id_1)(cos(bt)+isin(bt))+(c_2+id_2)(cos(bt)-isin(bt))] = [\text{Now, distribute the constants over the trigonometric functions}] y = e^{at}[(c_1cos(bt)+id_1cos(bt)+ic_1sin(bt)-d_1sin(bt)+c_2cos(bt)+id_2cos(bt)-ic_2sin(bt)+d_2sin(bt))]$

Next, group the real and imaginary parts:

$y = e^{at}[(c_1+c_2)cos(bt)+(d_2-d_1)sin(bt)]+ie^{at}[(c_1-c_2)sin(bt)+(d_1+d_2)cos(bt)]$

To ensure y is real, the imaginary part must vanish. This gives us the conditions: $c_1-c_2 = 0, d_1+d_2 = 0⇒ c_1 = c_2, d_1 = -d_2$. Thus, C₁ and C₂ must be conjugates. C₁ = c + id, C₂ = c -id.

$y = e^{at}[2ccos(bt)-2dsin(bt)] = e^{at}[c_1cos(bt)+c_2sin(bt)]$ where c₁ = 2c, c₂ = -2d, which matches the general form derived using the typical method Second-order Linear ODE’s with Constant Coefficients.

Undamped and Damped Oscillations

Understanding oscillatory systems is fundamental in maths, physics, and engineering. One of the simplest yet most illustrative models of oscillation is the mass-spring system. This system can exhibit various behaviors depending on parameters such as mass, spring stiffness, and damping. In this explanation, we’ll explore the mathematical modeling of the undamped and damped cases, solve their differential equations, and interpret the physical significance.

Mass-Spring System without Damping

Consider a mass-spring system where a mass m is attached to a spring with a spring constant k. The position of the mass at any time t is denoted by x(t), measured along the x-axis. We choose the origin such that the equilibrium position of the mass corresponds to x = 0. When displaced from equilibrium, the mass oscillates due to the spring’s restoring force.

Newton’s second law states that the sum of forces acting on the mass equals its mass times its acceleration: F_total = ma = mx’’ =[Notation] $m\frac{d^2x}{dt^2}$

In the undamped case (no friction or air resistance), the only force acting is the restoring force from the spring, described by Hooke’s Law: F_spring = -kx

Hooke’s law states that the force needed to extend or compress a spring by some distance x scales linearly (in other words, it is proportional to the displacement) with respect to that distance, F_spring = -kx where x is the displacement from equilibrium, and k is the stiffness of the spring. The negative sign indicates that the force opposes the displacement, aiming to restore the mass to equilibrium.

Setting F_total = F_spring: $m\frac{d^2x}{dt^2}=-kx$.

This leads to the second-order differential equation: $\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$. Define the natural angular frequency w₀ as: $w_0^2 = \frac{k}{m}$. The equation simplifies to: $\frac{d^2x}{dt^2} + w_0^2x = 0$. This is the standard second-order linear homogeneous ODE for simple harmonic motion.

Solution to the Undamped Equation

The characteristic equation is: $r^2 + w_0^2 = 0⇒[\text{Solving for r}] r = ± iw_0$. These are purely imaginary roots, corresponding to oscillatory behavior.

Second-order linear ODE’s. Complex roots. If the characteristic equation has complex roots, say r = a ± bi, the general solution is: y = $e^{at}(c_1cos(bt)+c_2sin(bt)) =$[Using the trigonometry identity a·cos(θ)+b·sin(θ) = c·cos(θ - Φ)] $e^{at}(c·cos(bt- Φ))$ where $c = \sqrt{c_1^2+c_2^2}$ and $Φ = arctan(\frac{c_2}{c_1})$.

The general solution is (a = 0, b = w₀): $x(t) = c_1cos(w_0t) + c_2sin(w_ot) = Acos(w_ot -Φ)$ where:

• A is the amplitude, A = $\sqrt{c_1^2+c_2^2}$.The maximum displacement from equilibrium.
• Φ is the phase shift, $Φ = arctan(\frac{c_2}{c_1})$
• w₀ is the natural angular frequency of oscillation.
• c₁ and c₂ are constant determined by initial conditions.

This solution represents simple harmonic motion, where the system oscillates indefinitely (undamped oscillations) with constant amplitude and frequency.

Inclusion of Damping

In real systems, oscillations are not eternal and damping cannot be ignored. Damping refers to the reduction of oscillation amplitude in a system over time due to energy loss. This energy loss can occur for several reasons, including: friction between moving parts, resistance in components (like resistors), or dashpots (like shock absorbers in vehicles).

The damping force F_damping ​is often proportional to the velocity and opposes the motion: F_damping = -cx’ =[Notation] $-c·\frac{dx}{dt}$ where:

• x’ = $\frac{dx}{dt}$ is the velocity.
• c is the damping coefficient, c > 0.

The negative sign indicates opposition to motion.

Total Force and Revised Equation of Motion

The total force on the mass is the sum of the spring force and the damping force. F_total = F_spring + F_damping = $-kx -c\frac{dx}{dt}$ (Refer to Figure 1 for a visual representation and aid in understanding it).

Applying Newton’s second law: $m\frac{d^2x}{dt^2} = -kx -c\frac{dx}{dt}$

Rewriting this equation gives the equation of motion: $m\frac{d^2x}{dt^2} +c\frac{dx}{dt} + kx = 0$ ↭ mx’’ + cx’ + kx = 0. This is a second-order linear differential equation that describes how the mass moves over time for a damped mass-spring system.

Standard Form of the Damped Oscillator Equation

$m\frac{d^2x}{dt^2} +c\frac{dx}{dt} + kx = 0$ ↭[Dividing through by m, we get the simplified form:] $\frac{d^2x}{dt^2} + \frac{c}{m}\frac{dx}{dt} + \frac{k}{m}x = 0$. Finally, it is typically expressed as $\frac{d^2x}{dt^2} + 2β\frac{dx}{dt} + w_o^2x = 0$ where:

• $2β = \frac{c}{m}$ is the damping factor (proportional to the damping coefficient). Damping refers to the process by which energy is lost in an oscillating system, usually due to friction, resistance or Dashpots, causing the amplitude of oscillations to decrease over time.

  Dashpots are commonly used to control motion and reduce oscillations, e.g.: shock absorbers in vehicles, door closers (they prevent doors from slamming shut), or industrial machinery (they reduce vibrations and noise in moving parts).

• $w_o^2 = \frac{k}{m}$ is the natural circular frequency squared (related to the stiffness and mass of the system).

Solving the Damped Oscillator Equation

$\frac{d^2x}{dt^2} + 2β\frac{dx}{dt} + w_o^2x = 0$ is the standard form of the equation for a damped mass-spring system.

To solve this equation, we start by solving the characteristic equation. We assume a solution of the form y(t) = e^rt, where r is a constant. Plugging y(t) = e^rt into the differential equation gives: $r^2e^{rt}+2βre^{rt} + w_0^2e^{rt} = 0$. Dividing through e^rt (which is never zero), we get the characteristic equation.

The characteristic equation associated with this differential equation is: r² + 2βr + w₀² = 0.

This is a quadratic equation, and we can solve it using the quadratic formula: r = $\frac{-2β±\sqrt{4β^2-4w_0^2}}{2} = \frac{-2β±2\sqrt{β^2-w_0^2}}{2} = -β ± \sqrt{β^2-w_0^2}$

Now, the behavior of the system, i.e., the nature of the solution depends on the discriminant Δ = $β^2-w_0^2$:

1. Overdamped: β > ω₀ (real and distinct roots).
2. Critically Damped: β = ω₀ (repeated real root).
3. Underdamped: β < ω₀ (complex conjugate roots).

Different Cases Based on Damping

Case 1: Overdamped (β > w₀)

When $β > w_0⇒ β^2 > w_0^2$, the discriminant is positive Δ = $4(β^2 - w_0^2)$ > 0, the characteristic equation has two distinct real roots: $r_{1, 2} = -β ± \sqrt{β^2-w_0^2}$.

Second-order linear ODE’s. Real and distinct roots. In this case, the solution is a combination of two exponentially decaying terms: y(t) = $c_1e^{r_1t} + c_2e^{r_2t} = c_1e^{(-β + \sqrt{β^2-w_0^2})t}+c_2e^{(-β - \sqrt{β^2-w_0^2})t}$. Since r₁ and r₂ are negative, this describes an overdamped motion, where the system does not oscillate but instead slowly returns to equilibrium.

Case 2: Critically Damped (β = w₀)

When $β^2 = w_0^2$ ⇒ Δ = 0, the system is critically damped, meaning it returns to equilibrium as quickly as possible without oscillating. The characteristic equation has a repeated real root: r = -β

Second-order linear ODE’s. Two equal roots, critically damped ODE. The general solution takes the form: y(t) = (c₁ + c₂t)e^-βt. This is the case of critical damping, where the system returns to equilibrium position without oscillating, but it does so as fast as possible. This occurs in systems where the damping is precisely balanced to prevent oscillations. It is often desired in systems like door closers or vehicle suspensions.

Case 3: Underdamped Oscillations (β < w₀)

In this case, when the damping is less than the circular frequency, $β^2 < w_0^2$, the discriminant is negative Δ < 0 and the roots of the characteristic equation are complex conjugates: $r = -β ± i\sqrt{w_0^2-β^2} = -β ± i\sqrt{w_d^2} = -β ± iw_d$ where $w_d = \sqrt{w_0^2-β^2}$.

Second-order linear ODE’s. Complex roots. This leads to a general solution involving both oscillatory and exponential decay terms:

y(t) = $e^{-βt}(c_1cos(\sqrt{w_0^2-β^2}t)+c_2sin(\sqrt{w_0^2-β^2}t)) = e^{-βt}(c_1cos(w_dt)+c_2sin(w_dt)) = e^{-βt}Acos(w_dt-Φ)$ where:

• $cos(\sqrt{w_0^2-β^2}t)$ and $sin(\sqrt{w_0^2-β^2}t)$ represent the oscillatory motion. The motion oscillates with a frequency w_d, called the damped angular frequency: $w_d = \sqrt{w_0^2-β^2}$.
• β is the damping. e^-βt represents the damping factor (exponential decay), which reduces the amplitude of the oscillations over time due to energy loss (e.g., friction between moving parts, air resistance, dashpots, such as shock absorbers in vehicle). If β = 0 (no damping) means the system conserve energy and oscillations continue indefinitely (simple harmonic motion).
• A and Φ depends on the initial conditions. $A=\sqrt{c_1^2+c_2^2}$ is the amplitude, Φ $= arctan(\frac{c_2}{c_1})$ is the phase angle.

This type of solution describes underdamped oscillations, where the system still oscillates, but with decreasing amplitude. The system loses energy due to damping, (refer to Figure 4 for a visual representation and aid in understanding it). Underdamped motion is common in mechanical and electrical systems where some oscillation is tolerable.

If t₁ and t₂ are two moments where the solution crosses the x-axis, then $t_2 = t_1 + 2\frac{π}{w_d}$. The first time the solution crosses the x-axis is $w_dt_1-Φ = \frac{π}{2} ↭ cos(w_dt_1-Φ) = cos(\frac{π}{2}) = 0$; and the second time is $t_2 = t_1 + 2\frac{π}{w_d}↭w_dt_2 -Φ = w_d(t_1 + 2\frac{π}{w_d})-Φ = (w_dt_1-Φ) +w_d·2\frac{π}{w_d} = \frac{π}{2}+2π, cos(\frac{π}{2}+2π)=0$

Case 4: Undamped Oscillations (No Damping, β = 0)

When β = 0 (i.e., no damping), the damping coefficient c = 0, and the equation of motion reduces to $\frac{d^2x}{dt^2} +w_0^2x = 0$. This is the equation of simple harmonic motion, which describes pure oscillations without any energy loss.

It is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance or displacement of the object from the equilibrium position. The force acts towards the equilibrium position, in the opposite direction to the displacement. The object oscillates back and forth through the equilibrium position, i.e., the motion is described by a sinusoid.

Mathematically, the restoring force F is given by Hooke’s Law: F = -kx, where F is the restoring force, k is the spring constant (measures of stiffness), x is the displacement form equilibrium. The negative sign indicates the force is directed towards the equilibrium position.

The characteristic equation becomes r² + w₀² = 0.

Second-order linear ODE’s. Complex roots. Since the roots are purely imaginary: r = ± iw₀, the general solution involves sinusoidal functions (no exponential terms): y(t) = c₁cos(w₀t) + c₂sin(w₀t) = Acos(wt- Φ) where:

• w₀ = $\sqrt{\frac{k}{m}}$ is the natural circular frequency.
• A is the amplitude of the oscillations.
• Φ is the phase shift, determined by initial conditions.
• c₁ and c₂ are constants determined by initial conditions.

The solution describes an oscillatory motion with no damping. This type of oscillation occurs when there is no energy loss in the system (no friction or resistance) and the motion persists indefinitely with constant amplitude and frequency.

When β ≠ 0, the system experiences damping, which means that the oscillations will gradually decrease in amplitude over time due to energy loss.

Bibliography

1. NPTEL-NOC IITM, Introduction to Galois Theory.
2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts, Calculus and Calculus 3e (Apex). Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn, and MathMajor.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra.
8. MIT OpenCourseWare [18.03 Differential Equations, Spring 2006], YouTube by MIT OpenCourseWare.
9. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.