Homogeneous Linear Equations with Constant Coefficients & Cauchy-Euler Equations

1. The Second-Order Constant Coefficient Equation

Consider the homogeneous linear ODE with constant real coefficients:

Assuming a trial solution of exponential form $y(x) = e^{rx}$, we compute $y' = r e^{rx}$ and $y'' = r^2 e^{rx}$. Substituting into the ODE:

Since $e^{rx} \neq 0$ for all real $x$, $r$ must satisfy the characteristic (auxiliary) equation:

2. The Three Root Regimes

• Case 1: Real and Distinct Roots ($\Delta = b^2 - 4ac > 0$)
  The roots $r_1 \neq r_2$ are real numbers. Two linearly independent solutions are $y_1 = e^{r_1 x}$ and $y_2 = e^{r_2 x}$. The general solution is: $$y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x}$$
• Case 2: Real and Repeated Roots ($\Delta = b^2 - 4ac = 0$)
  There is a single root of multiplicity two: $r = -b / (2a)$. One solution is $y_1 = e^{rx}$. Using reduction of order, the second independent solution is $y_2 = x e^{rx}$. The general solution is: $$y(x) = (c_1 + c_2 x) e^{rx}$$
• Case 3: Complex Conjugate Roots ($\Delta = b^2 - 4ac < 0$)
  The roots are $r = \alpha \pm i \beta$, where $\alpha = -b/(2a)$ and $\beta = \frac{\sqrt{4ac - b^2}}{2a} > 0$. Using Euler's formula $e^{(\alpha \pm i\beta)x} = e^{\alpha x}(\cos\beta x \pm i \sin\beta x)$, the real linearly independent fundamental solutions are $y_1 = e^{\alpha x} \cos(\beta x)$ and $y_2 = e^{\alpha x} \sin(\beta x)$. The general solution is: $$y(x) = e^{\alpha x} \left[ c_1 \cos(\beta x) + c_2 \sin(\beta x) \right] = R e^{\alpha x} \cos(\beta x - \delta)$$

1. Generalization to $n$-th Order Equations

For an $n$-th order linear ODE $a_n y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_1 y' + a_0 y = 0$, the characteristic polynomial is:

By the Fundamental Theorem of Algebra, $P(r)$ has exactly $n$ complex roots (counting multiplicities):

• Each real root $r$ of multiplicity $k$ contributes $k$ linearly independent solutions: $$e^{rx}, \quad x e^{rx}, \quad x^2 e^{rx}, \quad \dots, \quad x^{k-1} e^{rx}$$
• Each complex conjugate pair $\alpha \pm i\beta$ of multiplicity $k$ contributes $2k$ linearly independent solutions: $$\begin{aligned} e^{\alpha x} \cos\beta x, \quad x e^{\alpha x} \cos\beta x, \quad \dots, \quad x^{k-1} e^{\alpha x} \cos\beta x \\ e^{\alpha x} \sin\beta x, \quad x e^{\alpha x} \sin\beta x, \quad \dots, \quad x^{k-1} e^{\alpha x} \sin\beta x \end{aligned}$$

The sum of all these solutions multiplied by arbitrary constants $c_1, \dots, c_n$ forms the complete general solution.

1. Standard Form of Cauchy-Euler Equations

A linear differential equation of the form:

where the power of $x$ matches the order of the derivative in each term, is called a Cauchy-Euler (or equidimensional) equation.

2. Second-Order Homogeneous Cauchy-Euler Equation

Consider $a x^2 y'' + b x y' + c y = 0$ for $x > 0$. We seek solutions of the form $y = x^m$. Differentiating:

Substituting into the ODE:

This yields the indicial (auxiliary) equation:

3. Three Cases of Solutions for Cauchy-Euler Equations

• Distinct Real Roots ($m_1 \neq m_2$): $y(x) = c_1 x^{m_1} + c_2 x^{m_2}$.
• Repeated Real Root ($m_1 = m_2 = m$): The second solution is obtained by logarithmic scaling: $$y(x) = x^m (c_1 + c_2 \ln x)$$
• Complex Conjugate Roots ($m = \alpha \pm i \beta$): Using $x^{\alpha \pm i\beta} = x^\alpha e^{\pm i \beta \ln x} = x^\alpha [\cos(\beta \ln x) \pm i \sin(\beta \ln x)]$: $$y(x) = x^\alpha \left[ c_1 \cos(\beta \ln x) + c_2 \sin(\beta \ln x) \right]$$