Higher Order Differentials

General Solution of Nonhomogeneous ODE

The general solution of a nonhomogeneous ODE is the sum of the complementary solution (yc) and a particular solution (yp): y = yc + yp.

Complementary Solution

The complementary solution (y_c) is the general solution of the associated homogeneous ODE.

Particular Solution

A particular solution (y_p) is any solution to the nonhomogeneous ODE that contains no arbitrary constants.

Method of Undetermined Coefficients (MUC)

MUC is a method for finding y_p for nonhomogeneous linear ODEs with constant coefficients and specific forms of g(x).

MUC Applicability

MUC requires that the ODE is linear with constant coefficients and that g(x) produces only a finite number of linearly independent derivatives.

MUC Key Step

A key step in MUC is to modify the initial trial solution by multiplying by x's if any term in the trial solution duplicates a term in the complementary solution.

Method of Variation of Parameters (MVP)

MVP is a general method for finding y_p that can be used even when MUC is not applicable.

MVP Formula (2nd Order)

For a 2nd order ODE y''+p(x)y'+q(x)y=r(x), the particular solution is y_p = v1y1 + v2y2, where v1' = W1/W and v2' = W2/W, and W is the Wronskian of y1 and y2.

Wronskian Definition

The Wronskian of two functions y1 and y2 is W(y1, y2) = y1y2' - y2y1'.

Cauchy-Euler Equation Form

A Cauchy-Euler equation has the form ax²y'' + bxy' + cy = 0, where the coefficients are powers of x.

Cauchy-Euler Solution Method

To solve a homogeneous Cauchy-Euler equation, assume a solution of the form y = x'm, leading to a characteristic equation for m.

Spring-Mass System ODE

The general ODE for a spring-mass system is mu'' + cu' + ku = F(t), where m is mass, c is damping, k is stiffness, and F(t) is external force.

Free Oscillation

Free oscillation occurs when there is no external force acting on the system, F(t) = 0.

Forced Oscillation

Forced oscillation occurs when an external force F(t) ≠ 0 is applied to the system.

Undamped Free Oscillation

For an undamped free system (c=0, F(t)=0), the solution is sinusoidal: u(t) = A cos(ωt) + B sin(ωt), where ω = √(k/m) is the natural frequency.

Damping Coefficient

The damping coefficient c determines the type of damping in a system.

Critical Damping

Critical damping occurs when c = c_cr = √(4mk) = 2√(mk). The system returns to equilibrium fastest without oscillating.

Overdamped System

An overdamped system occurs when c > c_cr. The solution is a sum of real exponentials, and the system returns to equilibrium without oscillating, but slower than critical.

Underdamped System

An underdamped system occurs when c < c_cr.

Resonance

Resonance occurs in a forced, undamped system when the frequency of the external force (Ω) equals the natural frequency of the system (ω).

Resonance Solution Form

For an undamped system at resonance, the particular solution contains a term that grows linearly with time, like (F₀/(2mω)) t sin(ωt).

Transient and Steady-State

The solution of a forced, damped system consists of a transient component (from yc, which decays to zero) and a steady-state component (from yp, which persists).

Principle of Superposition

For a linear homogeneous ODE, any linear combination of solutions is also a solution.

Linearly Independent Functions

A set of functions is linearly independent if no function in the set can be written as a linear combination of the others.

Characteristic Equation

For a linear homogeneous ODE with constant coefficients, the characteristic equation is obtained by substituting y = e'(λt) and solving for λ.

Complex Conjugate Roots

If the characteristic equation has complex conjugate roots α ± βi, then the corresponding part of y_c is e'(αt)(c1 cos(βt) + c2 sin(βt)).

Static Equilibrium

In a spring-mass system, static equilibrium is the position where the force of gravity is balanced by the spring force, mg = kL.

Dashpot Force

A dashpot provides a damping force Fd that is proportional to velocity, Fd = c v, and opposes the direction of motion.

Natural Frequency

The natural frequency of an undamped spring-mass system is ω = √(k/m).

Beats Phenomenon

Beats occur in forced, undamped oscillations when the forcing frequency Ω is close to, but not equal to, the natural frequency ω, resulting in a periodic variation in amplitude.