Euler Equations and Order Reduction

Special Types of Linear ODEs

Euler Equation: Variable Coefficients of a Special Kind

Euler equation: $x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \ldots + a_1 x y' + a_0 y = f(x)$.

Feature: the coefficients of $y^{(k)}$ are proportional to $x^k$. This is not accidental—such equations arise when solving problems with natural radial symmetry (polar, cylindrical, spherical coordinates), as well as in the search for power solutions of more general equations.

Substitution: $t = \ln x$ (for $x > 0$), that is, $x = e^t$. Let $D = \frac{d}{dt}$.

Key formulas: $x \frac{dy}{dx} = D y$, $x^2 \frac{d^2y}{dx^2} = D(D-1) y$, $x^3 \frac{d^3y}{dx^3} = D(D-1)(D-2) y$, and in general $x^k y^{(k)} = D(D-1)\cdots(D-k+1) y$.

After substitution, the equation turns into a linear ODE with constant coefficients with respect to $t$—which we know how to solve!

Detailed Example: Euler Equation of Second Order

Equation: $x^2y'' - 3x y' + 4y = 0$.

Substitution $t = \ln x$: $x y' = D y$, $x^2 y'' = D(D-1) y$. Equation: $D(D-1)y - 3D y + 4y = 0$, that is, $(D^2 - 4D + 4)y = 0$.

Characteristic: $\lambda^2 - 4\lambda + 4 = 0 \rightarrow (\lambda-2)^2 = 0 \rightarrow \lambda = 2$ (multiplicity 2).

General solution in $t$: $y = (C_1 + C_2 t) e^{2t}$.

Return to $x$: $t = \ln x$, $e^{2t} = e^{2 \ln x} = x^2$. Answer: $y = (C_1 + C_2 \ln x) x^2$.

Verification: Compute $y' = 2C_1 x + C_2 (1 + 2 \ln x)x$ and $y'' = 2C_1 + C_2(3 + 4 \ln x)$... (left to the reader).

Power Series Solutions: The Frobenius Method

For the equation $x^2 y'' + x P(x) y' + Q(x) y = 0$, where $P(x)$ and $Q(x)$ are analytic near zero, we seek the solution in the form of a power series $y = x^r \sum a_n x^n$.

The exponent $r$ is determined from the indicial equation: $r(r-1) + P(0) r + Q(0) = 0$.

The Frobenius method allows us to construct solutions of Bessel's equation, Legendre's equation, the hypergeometric equation—all classic equations of mathematical physics.

Order Reduction

If one solution $y_1$ of a homogeneous equation of order $n$ is known, the order of the equation can be reduced by 1. The idea: substitution $y = y_1 \cdot v(x)$.

For a second-order equation: $y_1 v'' + 2 y_1' v' + y_1'' v + p(y_1 v' + y_1' v) + q y_1 v = 0$. Since $L[y_1] = y_1'' + p y_1' + q y_1 = 0$, terms with $v$ drop out. What remains: $y_1 v'' + (2 y_1' + p y_1) v' = 0$.

Substitution $w = v'$ lowers the order: $y_1 w' + (2 y_1' + p y_1) w = 0$—this is a linear first-order equation in $w$!

Example: Find the second solution of $y'' - y'/x + y/x^2 = 0$, if $y_1 = x$.

Substitute $y = x v$: $x v'' + 2v' - v'/x + v' - x v / x^2 \cdot x + ...$ Simplifying: $x v'' + (2 - 1) v' = x v'' + v' = 0$. Substitution $w = v'$: $x w' + w = 0 \rightarrow (x w)' = 0 \rightarrow w = C/x \rightarrow v = C \ln x$. Second solution: $y_2 = x \ln x$.

Sturm–Liouville Problem

Boundary value problem: $(p(x) y')' + (\lambda q(x) - r(x)) y = 0$ on $[a, b]$ with boundary conditions.

Unlike the Cauchy problem (initial conditions at one point), boundary conditions are given in two points. This fundamentally changes the nature of the problem: a solution exists only for special values of the parameter $\lambda$—eigenvalues.

Theorem: Under regular conditions, there exists a countable system of eigenvalues $\lambda_1 < \lambda_2 < \lambda_3 < \ldots \rightarrow +\infty$ with corresponding eigenfunctions $y_1, y_2, y_3, \ldots$

Orthogonality: $\int_a^b y_m(x) y_n(x) q(x) dx = 0$ for $m \neq n$.

Physical significance: Eigenvalues are the admissible frequencies of oscillation of the system (e.g., normal modes of vibration of a string). Eigenfunctions are the profiles of these oscillations. An arbitrary initial deviation can be expanded in eigenfunctions (the Fourier method for partial differential equations).

Example — string: $y'' + \lambda y = 0$, $y(0) = y(L) = 0$. Eigenvalues: $\lambda_n = (n\pi/L)^2$, $n = 1, 2, 3, \ldots$ Eigenfunctions: $y_n = \sin(n\pi x / L)$. These are precisely the harmonics of the string, discovered by Pythagoras!

Bessel Equation in Acoustics and Electromagnetism

Bessel equation $x^2 y'' + x y' + (x^2 - \nu^2) y = 0$ arises when separating variables in problems with cylindrical symmetry. Acoustics: vibrations of air in a round tube. Eigenfrequencies are the zeros of Bessel functions $J_n(k a)$, where $a$ is the radius of the tube. Electromagnetism: TM and TE modes in a cylindrical waveguide. Heat conduction: stationary temperature field in a cylinder is described by Bessel's equation after switching to cylindrical coordinates. Quantum mechanics: wave functions of the hydrogen atom in spherical symmetry—Legendre polynomials (angular part) and Bessel functions of half-integer order (radial part). Tables of zeros of Bessel functions $J_0, J_1$ are a standard tool for acoustical and radio engineers.

Question for reflection: Why does the Frobenius method yield a solution in the form of a power series, and not a closed formula? In which physical problems do the Bessel and Legendre equations arise naturally?