Differential Equations Reading Questions

Explain what a differential equation is using your own words.

An equation that relates a function with its derivatives.

What does it mean to be a solution to a differential equation?

A function that satisfies the equation when substituted into it.

What is a separable differential equation?

A type of differential equation that can be expressed as the product of a function of the independent variable and a function of the dependent variable.

N(y) dy/dx = M(x)

Explain why solution curves to a differential equation cannot intersect.

Graphs of solutions to a differential equation, which cannot intersect due to uniqueness of solutions theorem.

We can use Taylor polynomials to approximate a function f (x) near a point x₀. Explain why this approximation can only be expected to be accurate near x_0.

Polynomials used to approximate functions near a specific point, accurate primarily close to that point. The Taylor polynomial does not model f(x) accurately for values far from x₀.

What important rule from differential calculus do we use when solving a first-order differential equation?

Use the Produce Rule to solve first-order differential equations after multiplying by an integrating factor.

What is a first-order linear differential equation?

A differential equation that involves only the first derivative of the unknown function.

x’ + p(t) * x = q(t)

Explain what a bifurcation is in your own words.

A change in the number or stability of solutions of a differential equation as a parameter changes.

What is the characteristic equation of ax′′ + bx′ + cx = 0?

ar² + br + c = 0

Suppose that xp(t) and xq(t) are two solutions of ax′′ + bx′ + cx = g(t). How are these two solutions related?

x = xp(t) - xq(t)

Describe the Method of Undetermined Coefficients

Guessing a form of the solution based on the forcing function g(t), substituting the guess into the differential equation, then solving for the coefficients to find the particular solution.

Complexification

Using Euler’s formula e^iBt=cos(Bt) + isin(Bt) to derive a particular solution with the form x_c = x_Re + ix_Im

Complexified version z(t)

The relationship between x(t) and z(t) is that x(t) = Re(z(t))

Resonance

A phenomenon that occurs when a system oscillates at an increased amplitude due to matching of the system's natural frequency with the forcing frequency of an undamped harmonic oscillator.

Future behavior of bridge with behavior x’’ + 4x = 4sin(2t), x(0) = x’(0) = 0

The midpoint of the bridge starts at equilibrium.

As t increases, the bridge will oscillate with an increasing amplitude.

Heaviside Function

A step function/discontinuous function used to represent sudden changes in a system. The function outputs 0 for negative inputs and 1 for positive inputs

u_c(t) =

0 when t < c

1 when t >= c

Laplace Transform for IVPs

Use initial conditions to find the Laplace Transform of both sides of the equation ay’’ + by’ +cy = f(t), then find the inverse Laplace Transform of Y to find the solution.

Impulse Forcing

A sudden force applied to a system, such as a hammer strike, used to analyze transient responses.

Described by the Dirac delta function.

Eigenvalue

A scalar indicating how much a corresponding eigenvector is stretched or shrunk during a linear transformation.

Found by solving for the roots of the characteristic polynomial det[A-lambda * I) = 0

Eigenvector

A non-zero vector that only changes by a scalar factor when a linear transformation is applied.

Found by solving for x, y

(x; y)(A-lambda*I) = (0; 0)

Linearly Independent Vectors

Vectors that cannot be expressed as a linear combination of each other. They are not multiples of each other and do not lie on the same line through the origin.

Principle of Superposition for linear systems of differential equations

Any linear combination of solutions to the linear system x’ = Ax is also a solution.

Straight-line Solution

A point on a straight line where the vector field points in the same direction as the vector from the origin to the point.

stable line of solutions if all solutions approach (0,0)

unstable line if all nonzero solutions approach infinity

Distinct Real Eigenvalues Phase Plane

saddle: one negative, one positive

sink: both negative

source: both positive

Complex Eigenvalues Phase Plane

spiral sink: a < 0

spiral source: a > 0

center: a=0

Euler's Formula

A formula that connects complex exponentials and trigonometric functions, expressed as e^(ix) = cos(x) + i*sin(x).

Repeated Eigenvalues

A situation where a linear system has eigenvalues of the same value, leading to a single line of solutions.

2×2 linear system: Possible Phase Planes near the Origin

center, spiral sink, spiral source

3x3 Linear System Stability

All solutions of the system with initial conditions on the stable line approach the origin

All solutions with initial conditions that are non-zero points on the unstable plane would move away from the origin

Exponential of a Matrix A

A matrix function which generalizes the matrix exponential, used to solve systems of linear differential equations.

e^A = Sigma A^n / n!

Linearize a Nonlinear System

To approximate a nonlinear system by a linear system around a particular point.

Linear Part of a System of Equations

dx/dt = f(x,y)

dy/dt = g(x,y)

The portion of the system that involves only the first-degree terms of the variables that is a good approximation of f(x,y) and g(x,y)

Hamiltonian System

dx/dt = Hy

dy/dt = -Hx

Smooth function

Only saddles and centers are possible.

Constant Quantity in Hamiltonian System

The Hamiltonian itself (H) is conserved along any solution curve of the system.