Basics of Differential Equations - Flashcard set - Dynamical Systems

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Legendre's Differential Equation

Solution Approach: Use Legendre Polynomials. Example:

(1 - x^2)y'' - 2xy' + n(n+1)y = 0

The solution involves Legendre polynomials

P_n(x)

for non-negative integer

n

.

Bessel's Differential Equation

Solution Approach: Use Bessel functions of the first and second kind. Example:

x^2y'' + xy' + (x^2 - n^2)y = 0

The solution is

y(x) = C_1J_n(x) + C_2Y_n(x)

, where

J_n

and

Y_n

are Bessel functions.

Second-Order Constant Coefficient Homogeneous ODE

Solution Approach: Solve the characteristic quadratic equation. Example:

y'' + ay' + by = 0

Solve the characteristic equation

r^2 + ar + b = 0

to find the general solution.

Autonomous Differential Equation

Solution Approach: Analyze critical points and phase lines. Example:

\frac{dx}{dt} = f(x)

where

f(x)

does not depend on

t

. Study the signs of

f(x)

and solve for equilibrium solutions.

Partial Differential Equation (PDE)

Solution Approach: Several methods including separation of variables, transform methods, and numerical solutions. Example:

\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}

Choose an appropriate method based on type of PDE and boundary/initial conditions.

Second-Order Variable Coefficient Homogeneous ODE

Solution Approach: Use methods such as power series, Frobenius method, or numerical approximation. Example:

y'' + p(x)y' + q(x)y = 0

A specific analytic solution may not always be possible.

Separable Differential Equation

Solution Approach: Separate variables and integrate both sides. Example:

\frac{dy}{dx} = xy

Solve by separating variables and integrating:

\frac{1}{y} dy = x dx

Nonhomogeneous Linear Differential Equation

Solution Approach: Find the particular solution and add it to the homogeneous solution. Example:

ay'' + by' + cy = f(x)

Solve the homogeneous equation and then find a particular solution to

f(x)

.

Poisson's Equation

Solution Approach: Often solved using Green's functions or numerical methods. Example:

\nabla^2 u = f(x, y, z)

Where

f

is a given function. The specific approach depends on the domain and boundary conditions.

Fourier's Equation

Solution Approach: Similar to the heat equation, use separation of variables and Fourier series. Example:

u_t = \alpha^2 \nabla^2 u

Apply separation of variables and solve the resulting ordinary differential equations.

Heat Equation

Solution Approach: Use the Fourier series expansion for the initial condition. Example:

u_t = \alpha^2 u_{xx}

Where

\alpha

is the thermal diffusivity constant. The solution involves Fourier series and separation of variables.

First-Order Nonlinear Differential Equation

Solution Approach: Methods vary including graphical analysis, exact solution, or numerical methods. Example:

y' = f(y)

Solve analytically if possible, or use numerical solvers like Euler's method or Runge-Kutta method.

Coupled Differential Equations

Solution Approach: Reduce the system to a set of uncoupled equations. Example:

\begin{cases} \frac{dx}{dt} = f(x, y) \\ \frac{dy}{dt} = g(x, y) \end{cases}

Solve by diagonalization if linear, use substitutions or numerical methods if nonlinear.

Exact Differential Equation

Solution Approach: Verify it's exact, and then find the potential function. Example:

M(x,y)dx + N(x,y)dy = 0

Use the condition

\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}

to verify it's exact, then solve

\frac{\partial \Phi}{\partial x} = M

and

\frac{\partial \Phi}{\partial y} = N

.

Laplace's Equation

Solution Approach: Use separation of variables in multiple dimensions. Example:

\nabla^2 u = 0

Solve by assuming a solution of the form

u(x, y, z) = X(x)Y(y)Z(z)

and separate variables.

Linear First-Order Differential Equation

Solution Approach: Use an integrating factor. Example:

\frac{dy}{dx} + p(x)y = q(x)

Multiply through by the integrating factor

\mu(x)

to solve.

Homogeneous Linear Differential Equation

Solution Approach: Solve the characteristic equation. Example:

ay'' + by' + cy = 0

Find the roots of the characteristic equation

ar^2 + br + c = 0

to find the general solution.

Bernoulli Differential Equation

Solution Approach: Use a substitution to transform it into a linear equation. Example:

\frac{dy}{dx} + p(x)y = q(x)y^n

Make the substitution

u = y^{1-n}

and solve the resulting linear equation.

Riccati Differential Equation

Solution Approach: Requires a known particular solution or transformation to solve. Example:

y' = q_0(x) + q_1(x)y + q_2(x)y^2

If a particular solution

y_1(x)

is known, use the substitution

y = y_1(x) + \frac{1}{u}

.

Wave Equation

Solution Approach: Use D'Alembert's formula or separation of variables. Example:

u_{tt} = c^2 u_{xx}

Where

c

is the wave speed. For simple cases, use D'Alembert's formula. For problems with boundary conditions, use separation of variables.

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