Understanding Differential Equations

• Differential equations relate the variation of a quantity to other physical quantities with respect to time, space, or other parameters.
• Partial differential equations involve differentiation with respect to more than one variable.
• An example of a partial differential equation is Laplace's equation: V² V(r) = 0, which describes electrostatic potential V(r) in a charge-free region.
• A time-independent Schrödinger equation describes the dynamics of a particle also involves the Laplacian, also representing the kinetic energy of the particle, as well as a potential part.

Classifying Differential Equations

• Ordinary differential equations have unknown functions that depend on only one variable
• Partial differential equations have unknown functions dependent on more than one variable

Equation Characteristics

• Order: Defined by the highest derivative in the equation
• The order is still based on the highest derivative
• Linearity: A linear equation has no powers above the first power of the unknown function and its derivatives
• Homogeneity: A linear differential equation where all terms depend on the unknown function or its derivatives
• Degree: The exponent of the highest order derivative in the equation
• Only integer powers should be included

Solution Uniqueness and Existence

• Solutions: Direct substitution can verify if a function is a solution
• Uniqueness: Differential equations generally have more than one solution
• nth order equation typically has n independent functions as solutions
• Boundary conditions constrain the solution and determine unknown constants; n boundary conditions needed for an nth order equation

The Superposition Principle

• If V₁ and V₂ are solutions of a linear homogeneous differential equation, then a linear combination C₁V₁ + C₂V₂ is also a solution
• Under linearity and homogeneity assumptions, it's essential for this principle to hold
• The general solution includes a particular solution plus the general solution of the associated linear homogeneous equation

Separation of Variables

• Transforms an n-variable partial differential equation into n ordinary differential equations
• Assume a solution u(x, y) = X(x)Y(y) and separate variables.
• Separability: Essential to rewrite the equation so each side exclusively depends on one variable.
• Apply separation constants to turn the partial differential equations to ordinary

Laplace's Equation Example

• Laplace's equation (V²V = 0) can be used to determine the potential in Cartesian coordinates
• Assume a solution V(x, y, z) = X(x)Y(y)Z(z), then divide
• Introduce separation constants to get a set of ordinary differential equations: 1/X d²X/dx² = -l², 1/Y d²Y/dy² = -m², 1/Z d²Z/dz² = +n²
• Solutions involve trigonometric functions cos(lx), sin(lx) and exponential functions cosh(nz), sinh(nz)
• Apply boundary conditions to determine coefficients
• Superposition of separable solutions yields more general solutions

Spherical Coordinates

• Laplace’s equation is separable in spherical polar coordinates
• Expressed as:
• 1/r² ∂/∂r (r² ∂V/∂r) + 1/(r² sin θ) ∂/∂θ (sin θ ∂V/∂θ) + 1/(r² sin² θ) ∂²V/∂φ² = 0
• Separation results in 2 ODE -d/dr(r² dR/dr) = λR
• 1/sin θ d/dθ (sin θ dΘ/dθ) + (λ - m²/sin² θ) Θ = 0
• Solutions involve Legendre polynomials

One-Dimensional Wave Equation

• For a guitar string, the displacement y(x, t) follows: ∂²y/∂x² - 1/v² ∂²y/∂t² = 0
• Separation of variables y(x, t) = X(x)T(t) leads to two ODEs d²X/dx² + k²X = 0
• d²T/dt² + k²v²T = 0
• Applying boundary conditions (string clamped at x = 0 and x = L)
• y(0, t) = y(L, t) = 0
• Solution is a sum of sinusoidal functions with frequencies that are multiples of a fundamental frequency that creates the resonating sounds