Linear ODEs: Structure of the General Solution

Higher-Order Linear ODEs: Structure of the General Solution

Physical Motivation: Oscillations and Waves

Newton's law for a damped spring pendulum: $m\ddot{x} + c\dot{x} + kx = F(t)$. This is a second-order linear ODE with constant coefficients. The parameters $m$ (mass), $c$ (damping coefficient), and $k$ (spring stiffness) fully determine the system's behavior. Three qualitatively different regimes—oscillatory, transitional (critical), and aperiodic—are directly related to the type of roots of the characteristic equation. Understanding the structure of the solutions of this equation is the key to the entire theory of oscillations in physics and engineering.

General Theory of Linear ODEs

A linear ODE of order $n$: $ L[y] = y^{(n)} + p_{n-1}(x) y^{(n-1)} + ... + p_{1}(x) y' + p_{0}(x) y = f(x). $

The operator $L$ is linear: $L[c_1y_1 + c_2y_2] = c_1L[y_1] + c_2L[y_2]$.

Key theorem on the general solution: The general solution of the nonhomogeneous equation takes the form $y = y_p + y_g$, where $y_p$ is any particular solution of the nonhomogeneous equation, and $y_g$ is the general solution of the corresponding homogeneous equation $L[y] = 0$.

It follows that two problems must be solved: (1) find the solution space of the homogeneous equation; (2) find at least one particular solution of the nonhomogeneous equation.

The Wronskian Determinant

If $y_1, ..., y_n$ are $n$ solutions of the homogeneous equation, then the Wronskian determinant:

$ W(y_1, ..., y_n)(x) = \det [y_k^{(j-1)}(x)] $ for $k = 1, ..., n$ and $j = 1, ..., n$.

Abel's Theorem: $W(x) = W(x_0) \cdot \exp\left(-\int_{x_0}^x p_{n-1}(t) dt\right)$. This means: $W$ is either nowhere zero or identically zero. In the first case, the solutions are linearly independent (they form a fundamental system of solutions, FSS).

Practical significance: If the Wronskian is nonzero, all $n$ solutions "contribute independently" to the general solution—none of them can be expressed through the rest. It is precisely the FSS that forms the "basis" of the solution space.

Equations with Constant Coefficients

$ L[y] = y^{(n)} + a_{n-1} y^{(n-1)} + ... + a_{1} y' + a_{0} y = 0. $

Trial solution $y = e^{\lambda x}$: each derivative $y^{(k)} = \lambda^k e^{\lambda x }$. Substitute:

$ (\lambda^n + a_{n-1} \lambda^{n-1} + ... + a_1 \lambda + a_0) e^{\lambda x} = 0. $

Since $e^{\lambda x} \neq 0$, we obtain the characteristic equation: $P(\lambda) = \lambda^n + a_{n-1} \lambda^{n-1} + ... + a_0 = 0$.

Case 1. Distinct real roots $\lambda_1, ..., \lambda_n$. FSS = ${ e^{\lambda_1 x}, ..., e^{\lambda_n x} }$. General solution: $y = c_1 e^{\lambda_1 x} + ... + c_n e^{\lambda_n x}$.

Case 2. Multiple real root $\lambda$ of multiplicity $k$. Corresponding to it are $k$ linearly independent solutions: $e^{\lambda x},\ x e^{\lambda x},\ x^2 e^{\lambda x},\ ..., x^{k-1} e^{\lambda x}$.

Meaning: a multiple root "generates" a polynomial factor. This mathematically explains resonance phenomena in physics.

Case 3. Complex roots $\alpha \pm \beta i$. Real solutions: $e^{\alpha x} \cos(\beta x)$ and $e^{\alpha x} \sin(\beta x)$.

The Spring Pendulum: Full Analysis

Equation: $\ddot{x} + 2b\dot{x} + \omega_0^2 x = 0$, where $b = c/(2m)$ is the damping coefficient, $\omega_0^2 = k/m$ is the square of the natural frequency.

Characteristic equation: $\lambda^2 + 2b\lambda + \omega_0^2 = 0$. Roots: $\lambda = -b \pm \sqrt{b^2 - \omega_0^2}$.

Case $b < \omega_0$ (weak damping): Roots are complex: $\lambda = -b \pm i\omega$, where $\omega = \sqrt{\omega_0^2 - b^2}$. Solution: $ x(t) = e^{-bt} \left( C_1 \cos \omega t + C_2 \sin \omega t \right ) = A e^{-bt} \cos(\omega t + \varphi). $ Damped oscillations with amplitude $A e^{-bt}$ and frequency $\omega < \omega_0$.

Case $b = \omega_0$ (critical damping): Multiple root $\lambda = -b$. Solution: $ x(t) = (C_1 + C_2 t) e^{-bt}. $ Aperiodic return to equilibrium—the fastest return without overshooting zero.

Case $b > \omega_0$ (strong damping): Two distinct real roots $\lambda_1, \lambda_2 < 0$. Solution: $ x(t) = C_1 e^{\lambda_1 t} + C_2 e^{\lambda_2 t}. $ Aperiodic damping.

Example with numbers: Spring with $k = 4$ N/m, mass $m = 1$ kg, damping $c = 2$ Ns/m. Equation: $\ddot{x} + 2\dot{x} + 4x = 0$. Characteristic: $\lambda^2 + 2\lambda + 4 = 0$. Discriminant: $4 - 16 = -12 < 0$. Roots: $\lambda = -1 \pm i\sqrt{3}$. Solution: $ x(t) = e^{-t} (C_1 \cos(\sqrt{3} t) + C_2 \sin(\sqrt{3} t)). $ With initial conditions $x(0) = 0.1$ m, $\dot{x}(0) = 0$: $C_1 = 0.1,\ C_2 = 0.1/\sqrt{3} \approx 0.058$. Amplitude after $1$ s: $0.1 e^{-1} \approx 0.037$ m—decreased by a factor of $2.7$.

Question for reflection: Show that characteristic roots with negative real parts correspond to a stable equilibrium position of the spring, while those with positive real parts correspond to an unstable one. How does this relate to physical intuition?

Method of Variation of Parameters for Nonhomogeneous Equations

For the equation $y'' + p y' + q y = f(x)$ with a known fundamental set ${y_1, y_2}$, the method of variation of parameters seeks a solution $y = C_1(x)y_1 + C_2(x)y_2$. The condition $C_1' y_1 + C_2' y_2 = 0$ and $C_1' y_1' + C_2' y_2' = f$ give a system for $C_1'$ and $C_2'$ via the Wronskian determinant $W = y_1 y_2' - y_2 y_1'$. Result: $C_1' = -y_2 f / W$, $C_2' = y_1 f / W$. Advantage of the method: works for any right-hand side $f(x)$, even when the method of undetermined coefficients is not applicable (for example, for $f = \tan x$ or $f = e^x/x$, where the antiderivative cannot be expressed in elementary functions).