Euler Differential Equation — Definition, Formula & Examples

An Euler differential equation (also called a Cauchy–Euler equation) is a linear ODE where each coefficient is a power of the independent variable matching the order of its derivative. This structure allows it to be solved by substituting

y = x^r

and finding the values of

r

A Cauchy–Euler equation of order $n$ has the form $a_n x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \cdots + a_1 x\,y' + a_0\,y = 0$ , where the $a_k$ are real constants. The substitution $y = x^r$ (for $x > 0$ ) reduces it to an algebraic equation in $r$ , called the auxiliary (or indicial) equation.

Key Formula

a_2 x^2 y'' + a_1 x\,y' + a_0\,y = 0

Where:

$x$ = Independent variable (typically assumed $x > 0$)
$y$ = Unknown function of $x$
$a_0, a_1, a_2$ = Real constants (coefficients)

How It Works

To solve a Cauchy–Euler equation, assume a solution of the form

y = x^r

. Compute the necessary derivatives:

y' = r x^{r-1}

y'' = r(r-1)x^{r-2}

, and so on. Substitute these into the ODE; every term collapses to a constant times

x^r

, so you can factor out

x^r

and obtain a polynomial equation in

r

. If the roots of this auxiliary equation are real and distinct, the general solution is

y = C_1 x^{r_1} + C_2 x^{r_2}

. Repeated roots or complex roots require logarithmic or trigonometric modifications analogous to the constant-coefficient case.

Worked Example

Problem: Solve the Cauchy–Euler equation

x^2 y'' - 2x\,y' - 4y = 0

for

x > 0

Substitute y = x^r: Compute derivatives and substitute into the equation.

y = x^r,\quad y' = r x^{r-1},\quad y'' = r(r-1)x^{r-2}

Simplify: Each term becomes a constant times

x^r

. Factor out

x^r

x^r\bigl[r(r-1) - 2r - 4\bigr] = 0 \implies r^2 - 3r - 4 = 0

Solve the auxiliary equation: Factor or use the quadratic formula to find

r

(r - 4)(r + 1) = 0 \implies r_1 = 4,\; r_2 = -1

Write the general solution: Since the roots are real and distinct, combine the two independent solutions.

y = C_1 x^4 + C_2 x^{-1}

Answer:

y = C_1 x^4 + C_2 x^{-1}

, where

C_1

and

C_2

are arbitrary constants.

Why It Matters

Cauchy–Euler equations appear in problems with radial or scale-invariant geometry, such as heat conduction in cylindrical or spherical coordinates and stress analysis in elasticity. They also serve as a bridge to the method of Frobenius for solving ODEs with singular points.

Common Mistakes

Mistake: Forgetting that the power of

x

in each coefficient must match the order of the derivative it multiplies.

Correction: Before applying the

y = x^r

substitution, verify that the equation has the exact Cauchy–Euler form

a_k x^k y^{(k)}

for every term. If the powers of

x

do not match, the equation is not Cauchy–Euler and this method will not work directly.