ODE: Cauchy-Euler Equations

Mastering the Cauchy-Euler equation is a critical skill for engineers and applied scientists because it provides the mathematical toolkit for solving a broad class of variable-coefficient differential equations. These equations frequently appear in problems with geometric symmetry, such as analyzing stress in a thick-walled cylinder, modeling heat conduction in a wedge, or describing radial vibrations. While its variable coefficients might seem intimidating, the equation's structure allows for a powerful simplification through an intelligent change of variable, transforming it into a constant-coefficient problem we already know how to solve.

The Defining Form and the Core Insight

A Cauchy-Euler equation (also called an Euler-Cauchy or equidimensional equation) is a linear, homogeneous, second-order ordinary differential equation with variable coefficients that are specific powers of the independent variable. Its standard form is:

$$a{x}^2{y}^{\prime \prime }+bx{y}^{\prime }+cy=0,$$

where $a$, $b$, and $c$ are real constants, and we assume $x>0$ to avoid complications with logarithms and powers. The key observation is that each term has the same "dimension": if $y$ is dimensionless, then $x{y}^{\prime }$ and $x^2{y}^{\prime \prime }$ scale the same way. This suggests a solution of the form $y=x^r$, where $r$ is a constant to be determined. This is the first and most direct solution method.

Substituting $y=x^r$ into the ODE is straightforward. We compute the derivatives: $y^{\prime }=r{x}^{r-1}$ and $y^{\prime \prime }=r(r-1)x^{r-2}$. Plugging these into the standard form yields:

$$a{x}^2\cdot r(r-1)x^{r-2}+bx\cdot r{x}^{r-1}+c{x}^r=0.$$

Factoring out $x^r$ gives:

$$x^r[ar(r-1)+br+c]=0.$$

Since $x^r\ne 0$, we obtain the characteristic equation for Euler equations:

$$ar(r-1)+br+c=0\text{or, equivalently,}a{r}^2+(b-a)r+c=0.$$

This is an algebraic quadratic in $r$. The nature of its roots—distinct real, repeated real, or complex conjugate—directly dictates the form of the general solution, mirroring the process for constant-coefficient equations.

The Substitution Method: $x=e^t$

While the $y=x^r$ ansatz is effective, the substitution $x=e^t$ (or $t=\ln x$) provides a more systematic derivation and clearly shows the connection to constant-coefficient ODEs. This change of variable transforms the independent variable from $x$ to $t$. We then need to express the derivatives $y^{\prime }$ and $y^{\prime \prime }$ with respect to $x$ in terms of derivatives with respect to $t$.

Using the chain rule: $$\frac{dy}{dx}=\frac{dy}{dt}\cdot \frac{dt}{dx}=\frac{dy}{dt}\cdot \frac{1}{x}=e^{-t}\frac{dy}{dt}.$$

For the second derivative: $$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(e^{-t}\frac{dy}{dt}\right)=\frac{d}{dt}\left(e^{-t}\frac{dy}{dt}\right)\cdot \frac{dt}{dx}=e^{-2t}\left(\frac{d^{2}y}{d{t}^2}-\frac{dy}{dt}\right).$$

Substituting $x=e^t$, $x{y}^{\prime }=e^t\cdot (e^{-t}{y}_t)=y_t$, and $x^2{y}^{\prime \prime }=e^{2t}\cdot (e^{-2t}(y_{tt}-y_t))=y_{tt}-y_t$ (where subscripts denote derivatives with respect to $t$) transforms the original Cauchy-Euler equation into: $$a(y_{tt}-y_t)+b{y}_t+cy=0\Rightarrow a{y}_{tt}+(b-a)y_t+cy=0.$$

This is now a constant-coefficient linear ODE in the variable $t$! Its characteristic equation is $a{\lambda }^2+(b-a)\lambda +c=0$, whose roots $\lambda$ are identical to the roots $r$ found earlier. Solving this constant-coefficient equation and then substituting back $t=\ln x$ yields the solution in terms of $x$.

The Three Cases for the General Solution

The general solution to the homogeneous Cauchy-Euler equation is constructed from the roots $r_1$ and $r_2$ of its characteristic equation.

Case 1: Distinct Real Roots ($r_1\ne r_2$) The two linearly independent solutions are $x^{r_1}$ and $x^{r_2}$. The general solution is: $$y(x)=C_1{x}^{r_1}+C_2{x}^{r_2}.$$

Example: Solve $x^2{y}^{\prime \prime }-2x{y}^{\prime }-4y=0$. Assume $y=x^r$. The characteristic equation is $r(r-1)-2r-4=r^2-3r-4=(r-4)(r+1)=0$. Roots are $r_1=4$, $r_2=-1$. Thus, $y(x)=C_1{x}^4+C_2{x}^{-1}$.

Case 2: Repeated Real Root ($r_1=r_2$) When the characteristic equation has a double root $r_1$, one solution is $y_1=x^{r_1}$. A second, linearly independent solution is obtained by the reduction of order technique, yielding $y_2=x^{r_1}\ln x$. The general solution is: $$y(x)=x^{r_1}(C_1+C_2\ln x).$$

Case 3: Complex Conjugate Roots ($r=\alpha \pm i\beta$) If the roots are complex, the formal solutions are $x^{\alpha +i\beta }$ and $x^{\alpha -i\beta }$. Using Euler's formula, $x^{i\beta }=e^{i\beta \ln x}=\cos (\beta \ln x)+i\sin (\beta \ln x)$, we extract real-valued solutions. The general solution is: $$y(x)=x^{\alpha }[C_1\cos (\beta \ln x)+C_2\sin (\beta \ln x)].$$ The oscillatory behavior is now in terms of $\ln x$, not $x$, leading to phenomena like "log-periodic" oscillations common in scale-invariant systems.

Extending to Nonhomogeneous Cauchy-Euler Equations

Real-world applications often involve a forcing function. The nonhomogeneous Cauchy-Euler equation has the form: $$a{x}^2{y}^{\prime \prime }+bx{y}^{\prime }+cy=f(x).$$

The two primary methods for finding a particular solution ($y_p$) are variation of parameters and the method of undetermined coefficients—but the latter only works if $f(x)$ is a linear combination of terms of the form $x^k$, $x^k\ln x$, $(\ln x{)}^m$, or products of these with $\sin (\beta \ln x)$ or $\cos (\beta \ln x)$. This restriction exists because these are the forms that remain consistent under the $x=e^t$ substitution, where they become combinations of $e^{kt}$, $t{e}^{kt}$, $\sin (\beta t)$, etc., for which undetermined coefficients applies.

A more robust, systematic approach is to use the substitution $x=e^t$ first. This transforms the entire nonhomogeneous equation into: $$a{y}_{tt}+(b-a)y_t+cy=f(e^t)=g(t).$$ You then solve this constant-coefficient nonhomogeneous ODE for $y(t)$ using any standard method (undetermined coefficients for $g(t)$, variation of parameters, etc.). Finally, substitute back $t=\ln x$ to express the solution $y(x)$.

Applications to Problems with Radial Symmetry

The true power of the Cauchy-Euler equation is revealed in its applications to problems with radial symmetry. This is why it's indispensable in engineering fields like solid mechanics, fluid dynamics, and heat transfer.

Consider the classic example of plane strain or stress in a thick-walled cylindrical pressure vessel (like a pipe). Under axisymmetric loading, the governing equation for the radial displacement $u(r)$ often reduces to a Cauchy-Euler form: $$r^2\frac{d^{2}u}{d{r}^2}+r\frac{du}{dr}-u=0.$$ This is a Cauchy-Euler equation with $a=1$, $b=1$, $c=-1$. Its characteristic equation is $r(r-1)+r-1=r^2-1=0$, with roots $r=\pm 1$. The solution is $u(r)=C_{1}r+C_2/r$. The constants $C_1$ and $C_2$ are determined by the boundary conditions (e.g., internal and external pressures), leading directly to the Lamé equations for stress distribution.

Similarly, in heat conduction through a spherical or cylindrical shell with internal heat generation, the steady-state temperature profile $T(r)$ may satisfy a Cauchy-Euler type equation. The logarithmic and power-law solutions directly model how temperature and stress fields behave in these geometrically symmetric domains.

Common Pitfalls

1. Misapplying the Solution Form for $x<0$. The solutions $x^r$ and $\ln x$ are defined for $x>0$. For the interval $x<0$, replace every $x$ with $|x|$ in the final solution. For example, the general solution for repeated roots becomes $y(x)=|x{|}^{r_1}(C_1+C_2\ln |x|)$.

1. Forgetting the $\ln x$ Factor for Repeated Roots. A common error is to treat a repeated root $r_1$ as yielding two solutions $x^{r_1}$ and $x^{r_1}$ again. Remember, the second solution is always multiplied by $\ln x$ to ensure linear independence.

1. Incorrectly Handling Nonhomogeneous Equations with $x=e^t$. When using the $x=e^t$ substitution for a nonhomogeneous equation, students often forget to transform the forcing function $f(x)$ into $f(e^t)=g(t)$. You must work entirely in the $t$-domain before substituting back.

1. Misinterpreting Complex Root Solutions. The argument of the trigonometric functions is $\beta \ln x$, not $\beta x$. This means the oscillations occur more and more slowly as $x$ increases. Plotting the solution helps build intuition for this "log-periodic" behavior.

Summary

• The Cauchy-Euler equation $a{x}^2{y}^{\prime \prime }+bx{y}^{\prime }+cy=0$ is a variable-coefficient ODE solved by assuming a solution of the form $y=x^r$ or by using the transformative substitution $x=e^t$.
• Both methods lead to a characteristic equation $a{r}^2+(b-a)r+c=0$. The nature of its roots—distinct real, repeated real, or complex conjugate—dictates the form of the two linearly independent solutions, involving powers of $x$, $\ln x$, and trigonometric functions of $\ln x$.
• To solve nonhomogeneous Cauchy-Euler equations, the most systematic method is to first apply the substitution $x=e^t$ to convert it into a constant-coefficient ODE, solve it, and then convert back.
• These equations are fundamental for modeling phenomena with radial symmetry, such as stress in thick-walled cylinders and temperature distribution in spherical shells, making them an essential tool in mechanical, civil, and aerospace engineering analysis.