Solving Homogeneous Cauchy-Euler Differential

Equations
Forest W. Arnold
September 2018

1
Typeset in LATEX.

Copyright © 2018 Forest W. Arnold

This work is licensed under the Creative Commons Attribution-Noncommercial-Share

Alike 3.0 United States License. To view a copy of this license, visit
http://creativecommons.org/licenses/by-nc-sa/3.0/us/ or send a letter to
Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105,
USA. You can use, print, duplicate, share this work as much as you want. You can base
your own work on it and reuse parts if you keep the license the same.

Trademarks

TI-Nspire is a registered trademark of Texas Instruments, Inc.

2
1 Purpose
The objectives of this article are to
1. explain what Cauchy-Euler Equations are;

2. demonstrate how to find indicial equations for Cauchy-Euler Equations;

3. demonstrate how to solve Cauchy-Euler Equations using roots of indicial equa-
tions.

2 Cauchy-Euler Differential Equations

A Cauchy-Euler equation is a linear differential equation whose general form is

dny n−1 d
n−1 y dy
an x n + a n−1 x + · · · + a1 x + a0 y = g(x)
dxn dxn−1 dx
where an , an−1 , ... are real constants and an 6= 0.

The following paragraphs discuss solving second-order homogeneous Cauchy-Euler

equations of the form
d2y dy
ax2 2 + bx + cy = 0
dx dx
Note that when x = 0, ax2 = 0. Thus, when these equations are in standard form,
bx 0
y00 + ax c
2 y + ax2 y = 0, the equation has discontinuous coefficients at x = 0. The coeffi-
bx b c
cients ax 2 = ax and ax2 are not analytic at x = 0 (do not have Taylor series expansions
about x = 0) and x = 0 is a singular point. Unique solutions are guaranteed to exist
only in the intervals (−∞, 0) and (0, ∞).

TI-Nspire CAS’s deSolve() function for solving first- and second-order ordinary dif-
ferential equations does not solve these equations. Instead, it simply returns an equation
for y00 in terms of y0 , y, and x. The function de series sp() in the add-on library for
finding series approximations for differential equations, deserieslib.tns, can solve
many of these equations. However, it does not handle Cauchy-Euler equations with
complex solutions, solutions with complicated exponents, or equations with singular
points other than 0. Fortunately, these equations have closed-form solutions and are
easy to solve, even by hand.

3 Solution Method
2
Assume x > 0 and y = xr . Then y0 = dxd
(xr ) and y00 = dx
d r
2 (x ). Substituting these values
in the equation yields a quadratic equation multiplied by x : r

3
Since xr can never equal 0, both sides of this equation can be divided by xr , result-
ing in a simple quadratic equation called a characteristic or indicial equation:

Substituting xr for y in the differential equation and dividing both sides of the equa-
tion by xr transforms the equation to a quadratic equation in r. Therefore, when r is
a solution to the quadratic equation, y = xr is a solution to the differential equation.
Solving the differential equation requires finding the roots of a quadratic equation then
plugging those values into the correct solution form.

Solutions of quadratic equations are two roots, r1 and r2, which are either
1. real and unequal values, r1 6= r2,
2. real and equal values, r1 = r2, or
3. complex conjugates, α + β i, α − β i
When r1 6= r2, two linearly independent solutions of the equation are simply y1 = xr1
and y2 = xr2 and the general solution is y = c1xr1 + c2xr2 .

When r1 = r2, one independent solution is y1 = xr1 . A technique called reduction

of order is used to derive a second linearly independent solution, which is xr1 ln(x)
and the general solution is y = c1xr1 + c2xr1 ln(x).

When the roots are complex conjugates, one solution is y1 = xα+iβ and a second solu-
tion is y1 = xα−iβ . However, real-valued solutions are what are actually wanted. The
identity x = eln(x) and Euler’s formula is used to derive real-valued solutions:

xα+iβ = xα · xiβ = xα · eiβ ln(x) = xα [cos(β ln(x)) + sin(β ln(x))]

The two independent real-valued solutions are y1 = xα cos(β ln(x)) and y2 = xα sin(β ln(x))
and the general solution is y = xα [c1 cos(β ln(x)) + c2 sin(β ln(x)).

Solving second-order homogeneous Cauchy-Euler differential equations is achieved

with the following four simple steps:
1. Write the indicial equation using the constant coefficients a, b, c of the differen-
tial equation,

4
 2. Solve the indicial equation for its roots,
3. Determine the forms for the solution based on the type of roots of the indicial
equation, and
4. Write the solutions by plugging the roots in the solution form.
The following examples demonstrate how to solve these equations with TI-Nspire CAS
when x > 0.

4 Examples
4.1 Example 1
Solve the differential equation 3x2 y00 + xy0 − 8y = 0.

Step 1. For this equation, a = 3, b = 1, and c = −8. To write the indicial equation,
use the TI-Nspire CAS constraint operator to substitute the values of the constants in
the symbolic form of the indicial equation, indeqn = ar2 − (a − b)r + c = 0:

Step 2. Use the TI-Nspire CAS function cSolve() to find the roots of the indicial
equation:

Step 3. The roots are real and unequal, so the forms for the two independent solu-
tions are y1 = xr1 and y2 = xr2 .
4
Step 4. Let r1 = 2, r2 = − 34 . The independent solutions are y1 = x2 and y2 = x− 3
4
and the general solution is y = c1 · x2 + c2 · x− 3 .

4.2 Example 2
Solve the differential equation 4x2 y00 + y = 0.

Step 1. Write the indicial equation with a = 4, b = 0, c = 1.

5
Step 2. Find the roots of the indicial equation:

Step3. There is only one root, so r1 = r2 and the forms for the solutions are y1 = xr1
and y2 = xr1 · ln(x).
1 1
Step 4. Write the solutions using r1 = 21 : y1 = x 2 and y2 = x 2 · ln(x) and the gen-
1 1
eral solution is y = c1 · x 2 + c2 · x 2 · ln(x).

4.3 Example 3
Solve the differential equation 3x2 y00 + 5xy0 + y = 0.

Step 1. Write the indicial equation with a = 3, b = 5, c = 1:

Step 2. Find the roots of the indicial equation:

√ √
Step 3: The roots are complex conjugates, with r1 = − 31 + 32 i and r2 = − 13 − 32 i.
The forms for the solutions are y1 =√xα cos(β ln(x)) and y2 = xα sin(β ln(x)). From
the complex roots, α = − 31 and β = 32 .
1
√
Step 4: Write the solutions using the values of α and β : y1 = x− 3 cos( 3
2
ln(x))
√ √
− 31 − 13
and y2 = x sin( 32 ln(x)). The general solution is y = x [c1 · cos( 32 ln(x)) + c2 ·
√
sin( 32 ln(x))].

5 Solutions for x < 0

When x < 0, ln(x) has no real value (the value is complex). To solve Cauchy-Euler
differential equations for x real and x < 0, use the transformation ξ = −x, solve the
equation using ξ , then replace ξ with −x. However, since the indicial equation is iden-
tical for both x > 0 and x < 0, the transformations can be avoided by simply solving
the equation using x, then replacing x with its absolute value, |x|, in the solution. This
results in solutions for x 6= 0.

6
As an example, to solve Example 1, 3x2 y00 + xy0 − 8y = 0 when x < 0, first solve the
equation as above, then replace x with |x|. The solutions for x 6= 0 are y1 = |x|2 and
4 4
y2 = |x|− 3 and the general solution is y = c1 · |x|2 + c2 · |x|− 3 .

6 Solutions for x 6= x0
A more general form for a second-order homogeneous Cauchy-Euler equation is

d2y dy
a(x − x0 )2 + b(x − x0 ) + cy = 0
dx2 dx
The singular point for the general form is x = x0 instead of x = 0 (since x − x0 = 0 when
x = x0 ) and the solution intervals are (−∞ < x0 ) and (x0 < ∞).

The solution for the general form can be found by transforming (x − x0 ) to ξ , solv-
ing the equation, then transforming ξ back to (x − x0 ). The transformation can be
avoided since the indicial equation for the single variable, ξ , is the same as the indicial
equation for (x−x0 ) as shown by solving the equation 2(x+3)2 y00 +9(x+3)y0 −4y = 0:

Solving the indicial equation yields the two roots −4 and 12 :

Since there are two distinct roots, the two independent solutions for x 6= −3 are y1 =
1 1
|x + 3|−4 and y2 = |x + 3| 2 and the general solution is y = c1 · |x + 3|−4 + ·c2 · |x + 3| 2 .

7 Conclusion
Second-order homogeneous Cauchy-Euler differential equations are easy to solve. The
keys to solving these equations are knowing how to determine the indicial equation,
how to find its roots, and knowing which of the three forms for the solutions to use.
Although TI-Nspire CAS does not have a function for symbolically solving Cauchy-
Euler differential equations, its algebraic functions can minimize the effort required to
solve these differential equations.

7
References
[1] Nagle, R. Kent, Saff, Edward B., and Snider, Arthur David, Fundamentals of
Differential Equations and Boundary Value Problems, Pearson Education, Inc.,
2018, pp. 193-198

[2] Zill, Dennis G., A First Course in Differential Equations with Modeling Applic-
tions, 10th Ed., Brooks/Cole Cengage Learning, 2013, pp. 162-168
[3] Dawkins, Paul, ”Paul’s Online Notes”,http://tutorial.math.lamar.edu/
Classes/DE/EulerEquations.aspx