7.1 Second-Order Linear Equations

Second-order linear equations come in two flavors: homogeneous and nonhomogeneous. Homogeneous equations have no standalone x terms, while nonhomogeneous ones do. These equations are crucial for modeling real-world phenomena in physics, engineering, and other fields.

Solving these equations involves constructing characteristic equations and finding their roots. The nature of these roots determines the form of the solution. Initial or boundary conditions help pinpoint specific solutions, turning general solutions into particular ones that describe real-world scenarios.

Homogeneous and Nonhomogeneous Second-Order Linear Equations

Homogeneous vs nonhomogeneous equations

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• General form of a second-order linear equation: $y'' + p(x)y' + q(x)y = g(x)$
  • $p(x)$ and $q(x)$ are functions of the independent variable $x$ that multiply the first and second derivatives of $y$, respectively
  • $g(x)$ is a function representing the non-homogeneous term, which is a function of $x$ that does not involve $y$ or its derivatives
• Homogeneous second-order linear equation has $g(x) = 0$, meaning there is no term that is a function of only $x$ ($y'' + 2y' + 3y = 0$)
• Nonhomogeneous second-order linear equation has $g(x) \neq 0$, meaning there is a term that is a function of only $x$ ($y'' + 2y' + 3y = e^x$)
• These equations are examples of differential equations, which involve derivatives of an unknown function

Solving Homogeneous Second-Order Linear Equations

Construction of characteristic equations

• For a homogeneous equation $y'' + p(x)y' + q(x)y = 0$, assume a solution of the form $y = e^{rx}$, where $r$ is a constant to be determined
• Substitute $y = e^{rx}$, $y' = re^{rx}$, and $y'' = r^2e^{rx}$ into the homogeneous equation to obtain $r^2e^{rx} + p(x)re^{rx} + q(x)e^{rx} = 0$
• Divide by $e^{rx}$ to obtain the characteristic equation: $r^2 + p(x)r + q(x) = 0$, a quadratic equation in terms of $r$
• Solve the characteristic equation for the roots $r_1$ and $r_2$ using the quadratic formula or factoring

Solutions from characteristic roots

• Case 1: Distinct real roots ($r_1 \neq r_2$)
  • General solution: $y = c_1e^{r_1x} + c_2e^{r_2x}$, where $c_1$ and $c_2$ are arbitrary constants determined by initial or boundary conditions
• Case 2: Repeated real roots ($r_1 = r_2 = r$)
  • General solution: $y = (c_1 + c_2x)e^{rx}$, where the additional $x$ term accounts for the repeated root
• Case 3: Complex conjugate roots ($r_1 = \alpha + \beta i$, $r_2 = \alpha - \beta i$)
  • General solution: $y = e^{\alpha x}(c_1 \cos(\beta x) + c_2 \sin(\beta x))$, where the exponential term contains the real part $\alpha$ and the trigonometric terms contain the imaginary part $\beta$
• The solutions form a fundamental set of solutions, which are linearly independent

Solving Initial-Value and Boundary-Value Problems

Problems with boundary conditions

• Initial-value problem: Solve for the particular solution given initial conditions $y(x_0)$ and $y'(x_0)$ at a specific point $x_0$
  1. Substitute the initial conditions into the general solution and its derivative
  2. Solve the resulting system of equations for the constants $c_1$ and $c_2$
• Boundary-value problem: Solve for the particular solution given boundary conditions $y(x_1)$ and $y(x_2)$ at two different points $x_1$ and $x_2$
  1. Substitute the boundary conditions into the general solution
  2. Solve the resulting system of equations for the constants $c_1$ and $c_2$

General Solution and Initial Conditions

• The general solution of a second-order linear equation includes all possible solutions
• Initial conditions are used to determine the specific solution that satisfies given constraints
• The combination of the general solution and initial conditions yields a unique particular solution