Cauchy-Euler Equation

What is a Cauchy-Euler equation?

A Cauchy-Euler equation (also equidimensional or just Euler equation) is a linear ODE where each term has matching "dimensions" of $x$ and derivative, like $a_n \, x^{n} y^{(n)} + a_{n-1} \, x^{n-1} y^{(n-1)} + \ldots + a_1 \, x \, y' + a_0 \, y = 0.$

The second-order case: $a \, x^{2} y'' + b \, x \, y' + c \, y = 0.$

It is not constant-coefficient, but it has a similar clean structure because of the scaling symmetry: rescaling $x \to \lambda x$ leaves the equation form-invariant. This symmetry is why the characteristic ansatz $y = x^{r}$ works so cleanly here, and why "$e^{r t}$ for constant-coefficient ODEs" generalises to "$x^{r}$ for Cauchy-Euler ODEs".

Either ansatz is a special case of the observation: the solution space of a linear ODE with a continuous symmetry is spanned by functions that diagonalize that symmetry.

The given equation

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

Coefficients $a = 1$, $b = -2$, $c = 2$. Work on $x > 0$.

Step-by-step solution

Step 1 — Ansatz $y = x^{r}$.

Compute derivatives: $y = x^{r}, \qquad y' = r \, x^{r-1}, \qquad y'' = r(r-1) \, x^{r-2}.$

Plug into the ODE: $x^{2} \cdot r(r-1) x^{r-2} - 2 x \cdot r \, x^{r-1} + 2 \, x^{r}$ $= r(r-1) x^{r} - 2 r \, x^{r} + 2 \, x^{r} = x^{r}\bigl[r(r-1) - 2 r + 2\bigr]$ $= x^{r}\bigl[r^{2} - 3 r + 2\bigr].$

Step 2 — Characteristic (indicial) equation.

Since $x^{r} \ne 0$ on $x > 0$: $r^{2} - 3 r + 2 = 0 \implies (r - 1)(r - 2) = 0 \implies r_1 = 1, \, r_2 = 2.$

Step 3 — General solution.

Two distinct real roots → two power-law solutions: $\boxed{\, y(x) = C_1 \, x + C_2 \, x^{2} \,} \qquad (x > 0)$

Verification

$y = C_1 x + C_2 x^{2}$ $y' = C_1 + 2 C_2 x$ $y'' = 2 C_2$

$x^{2} y'' = 2 C_2 x^{2}$, $-2 x y' = -2 C_1 x - 4 C_2 x^{2}$, $2 y = 2 C_1 x + 2 C_2 x^{2}$.

Sum: $2 C_2 x^{2} - 2 C_1 x - 4 C_2 x^{2} + 2 C_1 x + 2 C_2 x^{2} = (2 - 4 + 2) C_2 x^{2} + (-2 + 2) C_1 x = 0$. ✓

The three cases of the indicial equation

Exactly like constant-coefficient ODEs, three flavors depending on the discriminant $\Delta = (b - a)^{2} - 4 a c$ (after rearranging the indicial equation $a r(r-1) + b r + c = 0$):

• Two real distinct roots $r_1 \ne r_2$: $y = C_1 x^{r_1} + C_2 x^{r_2}$. (This problem.)
• Repeated real root $r$: $y = x^{r}(C_1 + C_2 \ln x)$. The $\ln x$ replaces the $x$ factor we saw in constant-coefficient repeated-root cases (#185).
• Complex conjugate roots $r = \alpha \pm i \beta$: $y = x^{\alpha}\bigl(C_1 \cos(\beta \ln x) + C_2 \sin(\beta \ln x)\bigr)$. Oscillations in the logarithm of $x$ — the spacing between zeros gets tighter as $x \to 0^+$ or $x \to \infty$.

Mnemonic. Replace $e^{r t} \leftrightarrow x^{r}$ and $t \leftrightarrow \ln x$. The substitution $t = \ln x$ (so $x = e^{t}$) literally transforms a Cauchy-Euler ODE into a constant-coefficient one — see next section.

The substitution $t = \ln x$ (optional derivation)

Set $t = \ln x$, so $x = e^{t}$ and $dx = e^{t} dt = x \, dt$, i.e. $d/dx = (1/x) \, d/dt$.

$x \frac{dy}{dx} = \frac{dy}{dt}, \qquad x^{2} \frac{d^{2} y}{dx^{2}} = \frac{d^{2} y}{dt^{2}} - \frac{dy}{dt}.$

The Cauchy-Euler ODE $a x^{2} y'' + b x y' + c y = 0$ transforms to $a (y_{tt} - y_{t}) + b y_{t} + c y = 0 \implies a y_{tt} + (b - a) y_{t} + c y = 0,$ a constant-coefficient ODE in $t$. Solve it by the usual characteristic equation, then substitute $t = \ln x$ back.

For our problem: $y_{tt} + (-2 - 1) y_{t} + 2 y = 0 \implies y_{tt} - 3 y_{t} + 2 y = 0$, characteristic roots $r = 1, 2$, solutions $e^{t}, e^{2 t}$ → $x, x^{2}$. Matches step 3.

For $x < 0$

The ansatz $y = x^{r}$ is problematic for non-integer $r$ when $x < 0$. Use $y = |x|^{r}$ instead, or consider the two half-lines separately. Solutions on $x < 0$ are $C_1 |x| + C_2 |x|^{2}$ in this example.

Initial value problem

$y(1) = 3$, $y'(1) = 5$. From $y = C_1 x + C_2 x^{2}$, $y' = C_1 + 2 C_2 x$: $3 = C_1 + C_2, \quad 5 = C_1 + 2 C_2 \implies C_2 = 2, \; C_1 = 1.$ $y(x) = x + 2 x^{2}.$

Where Cauchy-Euler shows up

• Spherical or cylindrical coordinates with separation of variables. The radial part of Laplace's equation is often Cauchy-Euler.
• Scaling-invariant systems in physics and economics — anytime the law of the system is invariant under $x \to \lambda x$, expect power-law solutions.
• Boundary layer problems in fluid dynamics (inner scaling) often reduce to Cauchy-Euler form.

Common mistakes

• Using $e^{r x}$ ansatz for a Cauchy-Euler ODE. The right ansatz is $x^{r}$ — it's the scaling eigenfunction, not the translation eigenfunction.
• Forgetting the $-$ in $y(y - 1)$ when expanding $y'' = r(r-1) x^{r-2}$. It's the second derivative, not the first squared.
• Missing $\ln x$ for repeated roots. Analogous to the $x$ factor for repeated roots in constant-coefficient ODEs, but with $\ln x$ because of the log-substitution bridge.
• Applying at $x = 0$. The Cauchy-Euler equation has a singularity at the origin; solutions often behave badly there. Stick to $x > 0$ or $x < 0$.

Try it in the visualization

Slide the coefficients $a, b, c$ and watch the two indicial roots move. See the solution shape morph between power laws, logarithms (repeated root), and log-spaced oscillations (complex roots) — three qualitatively different regimes, all glued by the $x^{r}$ / $\ln x$ machinery.