Second-Order Linear Differential Equations

Study Notes

A differential equation is linear if it does not involve any powers, products, or other nonlinear functions of the dependent variable or its derivatives.
The general form of a second-order linear differential equation is: y" + P(x)y' + Q(x)y = R(x), where P, Q, and R are given functions of the independent variable x.
The function R(x) is called the nonhomogeneous term.
If R(x) = 0, the equation is homogeneous. Otherwise, it's nonhomogeneous.
A linear, nonhomogeneous equation often uses the related homogeneous equation (also called the complementary equation) for easier problem solving by setting R(x) to zero.

If functions P(x), Q(x), and R(x) are continuous on an interval X₁ < x < X₂, and x₀ is any point in this interval, then the differential equation y'' + P(x)y' + Q(x)y = R(x) has a unique solution. This solution satisfies the initial conditions: y(x₀) = y₀ and y'(x₀) = y'₀.

Two functions are linearly dependent in an interval if the ratio of the two functions is a constant. Otherwise, they are linearly independent.
The Wronskian of two functions y₁ and y₂ is denoted as W(y₁, y₂) and defined as W(y₁, y₂) = y₁y₂' - y₁'y₂.
Two functions are linearly independent if their Wronskian is never zero in the interval.

Superposition Principle: If y₁ and y₂ are two solutions of a linear homogeneous equation, then any linear combination y = C₁y₁ + C₂y₂ (where C₁ and C₂ are arbitrary constants) is also a solution of the equation.
If a function is a solution of a linear homogeneous differential equation, then any constant multiple of it is also a solution. If two functions are solutions of a linear homogeneous differential equation, then their sum is also a solution of that differential equation.

If a solution y₁ to a second-order linear homogeneous equation is known, a second linearly independent solution y₂ can be found in the form y₂ = v(x)y₁.
The method of reduction of order involves integration to find v(x). The general solution will be y = C₁y₁ + C₂y₂

The general solution of a second-order linear homogeneous differential equation with constant coefficients is typically of the form y = C₁e^(m₁x) + C₂e^(m₂x).
The values of m₁ and m₂ are determined from the characteristic equation. There are three possible root cases:
- real and distinct roots
- real and equal roots
- complex roots

If the roots (m₁, m₂) of the characteristic equation are real and unequal, the general solution is y = C₁e^(m₁x) + C₂e^(m₂x).

When the roots (m₁, m₂) are real and equal, the general solution is y = (C₁ + C₂x)e^(mx).

If the roots are complex conjugates (a ± iβ), the general solution becomes y = e^(ax)(C₁ cos(βx) + C₂ sin(βx)).

Superposition Principle for Particular Solutions: If y₁₂ is a particular solution to y" + P(x)y' + Q(x)y = R₁(x) and y₂p is a particular solution to y" + P(x)y' + Q(x)y = R₂(x), then y₁₂ + y₂p is a particular solution to y" + P(x)y' + Q(x)y = R₁(x) + R₂(x).

This is a technique for finding a particular solution (y_p) to a nonhomogeneous differential equation with constant coefficients.
It involves assuming a form for y_p based on the form of R(x), the nonhomogeneous term.
Common forms to assume are constants, polynomials, exponential functions, sine and cosine functions, and products of these.
Coefficients of the assumed form are then substituted and solved.

A general method for finding a particular solution to a nonhomogeneous linear second-order differential equation with variable or constant coefficients.
The approach relies on finding the general solution to the associated homogeneous equation, and then expressing the particular solution based on functions u₁(x) and u₂(x). These undetermined functions are derived through integration.