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

An algebraic equation is a mathematical statement that declares or asserts the equality of two algebraic expressions. These expressions are constructed using:

**Dependent and independent variables**. Variables represent unknown quantities. The independent variable is chosen freely, while the dependent variable changes in response to the independent variable.**Constants**. Fixed numerical values that do not change.**Algebraic operations**. Operations such as addition, subtraction, multiplication, division, exponentiation, and root extraction.

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, $\frac{dy}{dx}$

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 y’ = f(x, y) is continuous in a neighborhood around a point (x
_{0}, y_{0}) and - Its partial derivative with respect to y, $\frac{∂f}{∂y}$, is also continuous near (x
_{0}, y_{0}).

Then the differential equation y' = f(x, y) has a unique solution to the initial value problem through the point (x_{0}, y_{0}) .

A first-order linear differential equation (ODE) has the general form: a(x)y' + b(x)y = c(x) where y′ is the derivative of y with respect to x, and a(x), b(x), and c(x) are functions of x. If c(x) = 0, the equation is called homogeneous, i.e., a(x)y’ + b(x)y = 0.

The equation can also be written in the standard linear form as: y’ + p(x)y = q(x) where $p(x)=\frac{b(x)}{a(x)}\text{ and }q(x) = \frac{c(x)}{a(x)}$

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 $y=e^{(a+bi)t} = e^{at}e^{ibt} = e^{at}(cos(bt) + isin(bt))$⇒ 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 = $c_1y_1 + c_2y_2$. The constants may change, but we don't get any new results.

Now, consider the general solution of the form: $y = C_1e^{(a+bi)t}+C_2e^{(a-bi)t}$ where C_{1} and C_{2} are complex constants. To find real solutions, we need the imaginary parts of this expression to cancel out.

Let’s express C_{1} and C_{2} as complex numbers: C_{1} = c_{1} + id_{1} and C_{2} = c_{2} + id_{2} where c_{1}, c_{2}, d_{1}, d_{2} 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 have y be purely 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, C_{1} and C_{2} must be conjugates. C_{1} = c + id, C_{2} = c -id.

$y = e^{at}[2ccos(bt)-2dsin(bt)] = e^{at}[c_1cos(bt)+c_2sin(bt)]$ where c_{1} = 2c, c_{2} = -2d, which matches the general form derived using the typical method Second-order Linear ODE’s with Constant Coefficients.

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 factors such as mass, spring stiffness, and damping. In this explanation, we’ll explore the mathematical modeling of a damped mass-spring system, solve the resulting differential equations, and interpret the physical significance of these solutions.

Consider a mass-spring system where a mass 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 restoring force of the spring.

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] $\frac{d^2x}{dt^2}$**

In the undamped case (no friction or air resistance), the only force 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 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: $\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$. Let w_{0} be the natural angular frequency: $w_0^2 = \frac{k}{m}$. The equation simplifies to: $\frac{d^2x}{dt^2} + w_0^2x = 0$

The characteristic equation is: $r^2 + w_0^2 = 0⇒[\text{Solving for r}] r = ± iw_0$

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 = w_{0}): $x(t) = c_1cos(w_0t) + c_2sin(w_ot) = Acos(w_ot -Φ)$ where:

- A is the amplitude.
- Φ is the phase shift.
- c
_{1}and c_{2}are constant determined by initial conditions.

This solution represents simple harmonic motion: undamped oscillations with constant amplitude and frequency.

In real systems, damping cannot be ignored. Damping refers to the reduction of oscillation amplitude in a system, often 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.

The negative sign indicates opposition to 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.

$m\frac{d^2x}{dt^2} +c\frac{dx}{dt} + kx = 0$ ↭[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).

$\frac{d^2x}{dt^2} + 2β\frac{dx}{dt} + w_o^2x = 0$ 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: r^{2} + 2βr + w_{0}^{2} = 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 = $β^2-w_0^2$:

- Overdamped: β > ω
_{0}(real and distinct roots). - Critically Damped: β = ω
_{0}(repeated real root). - Underdamped: β < ω
_{0}(complex conjugate roots).

Case 1: **Overdamped (β > w _{0})**

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 r_{1} and r_{2} are negative, this describes **overdamped motion, where the system does not oscillate but instead slowly returns to equilibrium.**

Case 2: **Critically Damped (β = w _{0})**

When $β^2 = w_0^2$, 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) = (c_{1} + c_{2}t)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.

Case 3: **Underdamped Oscillations (β < w _{0})**

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: $r = -β ± i\sqrt{w_0^2-β^2} = -β ± i\sqrt{w_d^2} = -β ± iw_d$ 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: $w_d = \sqrt{w_0^2-p^2}$. - β is the damping. e
^{-βt}represents the damping factor (exponential decay), which reduces the amplitude of the oscillations over time. - A and Φ depends on the initial conditions.

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).

If t_{1} and t_{2} 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 $w_dt_2 -Φ = w_d(t_1 + 2\frac{π}{w_d})-Φ = (w_dt_1-Φ) +w_d·2\frac{π}{w_d} = \frac{π}{2}+2π$

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 r^{2} + w_{0}^{2} = 0.

Second-order linear ODE’s. Complex roots. Since the roots are purely imaginary: r = ± iw_{0}, the general solution involves sinusoidal functions (no exponential terms): y(t) = c_{1}cos(w_{0}t) + c_{2}sin(w_{0}t) = Acos(wt- Φ) where:

- w
_{0}= $\sqrt{\frac{k}{m}}$ is the natural circular frequency - A is the amplitude of the oscillation
- Φ is the phase shift, determined by initial conditions
- c
_{1}and c_{2}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.

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

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].

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