Euler--Cauchy and Higher-Order Equations

date2026-07-16document_iddoc_f18dc0e3974d69536d7995312dda8d2bdescriptionEuler-Cauchy 型方程式と高階線型定数係数方程式を、試行解の選択理由と根の重複から整理する。prerequisites二階線型定数係数微分方程式の拡張理論type講義content_typelecturestatusactive

mathdifferential-equationshigher-orderlecture

1Introduction

This lecture explains why exponential trial solutions apply to higher-order constant-coefficient equations and why power trial solutions apply to Euler--Cauchy equations.

2Higher-Order Constant-Coefficient Equations

For

a_{n} y^{(n)} + \dots + a_{1} y^{'} + a_{0} y = 0, a_{n} \neq 0,

the trial solution $y = e^{r x}$ produces the characteristic equation

a_{n} r^{n} + \dots + a_{1} r + a_{0} = 0.

If $r$ is a root of multiplicity $m$ , the associated solutions are

e^{r x}, x e^{r x}, \dots, x^{m - 1} e^{r x} .

These factors supply the number of independent solutions required by the order of the equation.

3Euler--Cauchy Equations

On an interval with $x > 0$ , an {Euler--Cauchy equation} has the form

x^{2} y^{'}' + a x y^{'} + b y = 0.

Substitution of $y = x^{r}$ gives $x y^{'} = r x^{r}$ and $x^{2} y^{'}' = r (r - 1) x^{r}$ , reducing the differential equation to an algebraic equation in $r$ .

This reduction follows systematically from the change of variable $x = e^{s}$ . If $D_{s} = d / d s$ , then

x y^{'} = D_{s} y, x^{2} y^{'}' = D_{s} (D_{s} - 1) y .

Thus the Euler--Cauchy equation becomes a constant-coefficient equation in $s = \log x$ , and $x^{r} = e^{r s}$ is its natural trial solution. On an interval with $x < 0$ , set $z = - x > 0$ and use $| x |^{r}$ and $\log | x |$ . Solutions must be treated separately on the two sides of the singular point $x = 0$ .

4Example

For

x^{2} y^{'}' + x y^{'} - y = 0,

substitution of $y = x^{r}$ gives

r (r - 1) + r - 1 = r^{2} - 1 = 0.

Hence $r = \pm 1$ and, on $x > 0$ ,

y = C_{1} x + C_{2} x^{- 1} .

5Repeated Roots and Logarithmic Factors

When the indicial equation has a repeated root, a second independent solution contains a logarithmic factor. For example,

x^{2} y^{'}' - x y^{'} + y = 0

gives $(r - 1)^{2} = 0$ and therefore

y = C_{1} x + C_{2} x \log x

on $x > 0$ . Under $s = \log x$ , the repeated-root solutions are $e^{s}$ and $s e^{s}$ ; transforming back gives $x$ and $x \log x$ .

6Connection with Higher-Order Theory

For higher-order constant-coefficient equations, the roots of the characteristic polynomial determine the elementary solution components. Distinct roots yield exponentials, while repeated roots introduce polynomial factors. This structure ensures that the dimension of the solution space equals the order of the equation.

7Scope and Limitations

The point $x = 0$ is singular for an Euler--Cauchy equation. A solution formula must therefore specify an interval contained either in $x > 0$ or in $x < 0$ ; one must not extend the same real formula across zero without justification. General variable-coefficient equations need not reduce to Euler--Cauchy form. Their analysis near singular points leads to the Frobenius method.

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