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Mathematics

Differential Equations

Bernoulli Differential Equations

Bessel's Differential Equations

Basic Differential Equations

Delay Differential Equations

Dynamical Systems and Phase Plane Analysis

First-Order Linear Differential Equations

Frobenius Method

Finite Difference Method for PDEs

Fourier Transform Methods for PDEs

Heat Equation

Laplace's Equation

Legendre's Differential Equations

Linear Second-Order Homogeneous Equations

Nonlinear Dynamics and Chaos

Perturbation Methods for Differential Equations

Poisson's Equation

Power Series Solutions of Differential Equations

Riccati Differential Equations

Sturm-Liouville Theory

Linear Second-Order Homogeneous Equations

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General form of a second-order homogeneous differential equation

Solving involves finding the roots of the characteristic equation, which can be real and distinct, repeated, or complex. The general solution will depend on the nature of the roots.

Wronskian and Linear Independence

The Wronskian determinant of two solutions of a second-order differential equation can be used to test for their linear independence, which is required for the general solution to span all solutions.

Characteristic equation with complex roots

The solution involves trigonometric functions:

y = e^{\alpha x}(C_1 \cos(\beta x) + C_2 \sin(\beta x))

where

\alpha

is the real part and

\pm \beta i

are the imaginary parts of the complex roots.

Cauchy-Euler Equations

A type of variable coefficient homogeneous second-order differential equation that can be transformed into a constant coefficient equation:

ax^2y'' + bxy' + cy = 0

solved by assuming a solution of the form

x^r

Homogeneous Equations with Variable Coefficients

These differential equations have no general method for solution but can sometimes be solved using power series, the method of Frobenius, or by recognizing a special form.

Reduction of Order

A technique used to find the second solution of a linear second-order differential equation when one solution is known: Assume the second solution has the form

v(x)y_1

and find

v(x)

by substitution into the original equation.

Characteristic equation with real and distinct roots

The solution is the sum of two exponential functions:

y = C_1e^{r_1x} + C_2e^{r_2x}

where

r_1

and

r_2

are the roots of the characteristic equation.

Superposition Principle

For linear homogeneous differential equations, the sum of two solutions is also a solution. Therefore, the general solution can be expressed as a linear combination of independent solutions.

Auxiliary or Characteristic Equation

The characteristic equation of a second-order homogeneous differential equation is

ar^2 + br + c = 0

which corresponds to the general form

ay'' + by' + cy = 0

when assuming solutions of the form

e^{rx}

Characteristic equation with repeated roots

The solution is an exponential function times a polynomial:

y = (C_1 + C_2x)e^{rx}

where

r

is the repeated root of the characteristic equation.

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