MATH250
Linear Equation: An equation where the variables appear to the first power and are not multiplied or divided by each other.
Linear Operator: A function that takes a function and produces another function, satisfying the properties of additivity and homogeneity.
Set of Fundamental Solutions of a Linear Equation: A set of solutions to a linear equation from which all other solutions can be formed by combining them.
Homogeneous Equation: An equation where everything equals zero, such as y′′+p(x)y′+q(x)y=0y'' + p(x)y' + q(x)y = 0y′′+p(x)y′+q(x)y=0.
General Solution: The most complete solution to a differential equation, which includes constants that can be adjusted based on initial conditions.
Particular Solution: A specific solution to a differential equation found by applying initial conditions or other constraints.
Characteristic Equation: An algebraic equation derived from a linear differential equation, used to find the roots that help form the solution.
Linearly Independent Functions: Functions that cannot be written as a combination of each other; no function in the set is redundant.
The Principle of Superposition: The idea that for a linear system, the sum of solutions is also a solution.
The Wronskian of Two Functions: A value that helps test if two functions are linearly independent. If it’s non-zero, the functions are independent.
Power Series: A sum of terms like a0+a1(x−c)+a2(x−c)2+...a_0 + a_1(x-c) + a_2(x-c)^2 + ...a0+a1(x−c)+a2(x−c)2+..., where each term is a power of (x−c)(x - c)(x−c).
Convergence of a Power Series: A power series converges if the sum of its terms gets closer to a specific value as you add more terms.
Center of a Power Series: The point ccc around which the series is expanded and evaluated.
Radius of Convergence: The distance from the center of the power series where the series converges.
Analytic Function: A function that can be written as a power series around any point in its domain.
Special Functions:
(a) A special function is: A function that has specific properties and appears frequently in various areas of mathematics and physics, often with well-defined solutions, such as the sine, cosine, or Bessel functions.
(b) Two examples of special functions are:
Bessel functions: Solutions to Bessel's differential equation, often used in problems with cylindrical symmetry.
Legendre polynomials: Solutions to Legendre's differential equation, commonly used in spherical problems.
(c) A function f(t)f(t)f(t) is analytic at t0t_0t0 if: It can be represented by a convergent power series around the point t0t_0t0, meaning it is differentiable in the neighborhood of t0t_0t0.
(d) A singular point of a function f(t)f(t)f(t) is: A point where the function is not analytic, meaning it cannot be expressed by a power series in the neighborhood of that point. This can occur due to a discontinuity or an undefined value.
(e) An ordinary point of a function f(t)f(t)f(t) is: A point where the function is analytic, meaning it has no singularities and can be expressed as a power series.
Continuity and Exponential Order:
(a) A function f(t)f(t)f(t) is piecewise continuous if: It is continuous on each piece of its domain, except possibly at a finite number of points where it may have jump discontinuities.
(b) A function f(t)f(t)f(t) has exponential order if: There exists a constant MMM and a value aaa such that ∣f(t)∣≤Meat|f(t)| \leq M e^{at}∣f(t)∣≤Meat for large values of ttt, meaning the function grows no faster than an exponential function as ttt increases.
(c) An improper integral ∫0∞f(t) dt\int_{0}^{\infty} f(t) \, dt∫0∞f(t)dt converges if: The limit of the integral exists and is finite, even if the upper limit of integration goes to infinity. This happens when the function approaches zero rapidly enough as t→∞t \to \inftyt→∞.
(d) A function f(t)f(t)f(t) has a Laplace transform if:
The function is piecewise continuous on [0,∞)[0, \infty)[0,∞).
The function has exponential order, meaning it grows no faster than an exponential function as t→∞t \to \inftyt→∞.