Cauchy-Euler Equations

Definition A second-order Cauchy-Euler equation, also called an equidimensional equation, is a linear equation

with real constants $a$ and $b$, where $y' = dy/dx$. An equation $\alpha x^2y'' + \beta xy' + \gamma y = 0$ with constants $\alpha \ne 0$, $\beta$ and $\gamma$ takes this form after division by $\alpha$. More generally, an equation of order $n$ is a Cauchy-Euler equation if it has the form

with constants $c_k$ and $c_n \ne 0$. The name refers to scaling. If $y$ solves the equation on an interval $I$ and $c \ne 0$, then $z(x) = y(cx)$ solves it for all $x$ with $cx$ in $I$, because

for every $k$.

Definition On the interval $x > 0$ the power $x^m = e^{m\ln x}$ is defined for every real number $m$, and

Substituting $y = x^m$ into the left side of $x^2y'' + a\,xy' + b\,y = 0$ gives $\bigl(m(m-1) + am + b\bigr)x^m$. Since $x^m \ne 0$, the power $x^m$ solves the equation on $(0, \infty)$ exactly when $m$ is a root of the power characteristic equation, also called the indicial equation,

If its roots $m_1 \ne m_2$ are real, the general solution on $(0, \infty)$ is $y = C_1x^{m_1} + C_2x^{m_2}$. The interval $(0, \infty)$ is a regular interval, because the standard-form coefficients $a/x$ and $b/x^2$ are continuous there, and the two powers form a fundamental set, because their Wronskian $(m_2 - m_1)\,x^{m_1 + m_2 - 1}$ is never zero there.

Definition On $x > 0$ set $\tau = \ln x$, so $x = e^{\tau}$, and let $Y(\tau) = y(e^{\tau})$ describe the same function in the new coordinate. Write $Y_\tau$ and $Y_{\tau\tau}$ for the first and second derivatives with respect to $\tau$. The chain rule gives

with the right sides evaluated at $x = e^{\tau}$. So $x^2y'' = Y_{\tau\tau} - Y_\tau$, and the Cauchy-Euler equation becomes the constant-coefficient equation

Its characteristic equation $r^2 + (a - 1)r + b = 0$ is the power characteristic equation, and a solution $e^{m\tau}$ becomes $e^{m\ln x} = x^m$. On $x < 0$ the same two identities hold with $\tau = \ln\lvert x \rvert$, so $x = -e^{\tau}$ and $Y(\tau) = y(-e^{\tau})$, and the solution formulas hold with $\lvert x \rvert$ in place of $x$. Neither coordinate reaches $x = 0$, so the two half-lines are solved separately. A right side $g(x)$ in a forced equation becomes $g(e^{\tau})$ on $x > 0$, so forced equations can be solved in $\tau$ as well.

Definition Let $a$ and $b$ be real, and let $m_1$ and $m_2$ be the roots of $m^2 + (a - 1)m + b = 0$. On $x > 0$ the general solution has one of three forms.

• Distinct real roots give $y = C_1x^{m_1} + C_2x^{m_2}$.
• A repeated root, which occurs when $(a - 1)^2 = 4b$ and equals $m = (1 - a)/2$, gives $y = x^m(C_1 + C_2\ln x)$.
• Complex roots $\alpha \pm i\beta$ with $\beta > 0$ give the real general solution below.

The factor $\ln x$ is the factor $\tau$ of the repeated-root solution $\tau e^{m\tau}$, and the complex case comes from the real and imaginary parts of $x^{\alpha + i\beta} = x^{\alpha}e^{i\beta\ln x}$. The pairs have Wronskians $(m_2 - m_1)\,x^{m_1 + m_2 - 1}$, $x^{2m - 1}$ and $\beta\,x^{2\alpha - 1}$, never zero on $x > 0$, so each pair is a fundamental set there. On $x < 0$ every formula holds with $\lvert x \rvert$ in place of $x$.

Definition The leading coefficient $x^2$ vanishes at $x = 0$, so the equation can be divided by it only for $x \ne 0$. In standard form

the coefficients are continuous on $(-\infty, 0)$ and on $(0, \infty)$. Each half-line is a regular interval, so data $y(x_0) = y_0$ and $y'(x_0) = y_1$ with $x_0 \ne 0$ determine exactly one solution on the half-line that contains $x_0$. When $a$ and $b$ are not both zero, at least one coefficient is unbounded near $0$, and the origin is a singular point. The existence and uniqueness theorem does not apply on any interval that contains $0$. A solution found on a half-line extends across $0$ only if the extended function is twice differentiable at $0$ and satisfies the original equation there, and data given at $x = 0$ may have no solution, exactly one, or infinitely many. In the language of series solutions this singular point is a regular singular point, the kind treated in Regular Singular Points and Frobenius.