Download Section 4.7

Math 3120
Differential Equations
with
Boundary Value Problems
Chapter 4:
Higher-Order Differential Equations
Section 4-7: Cauchy-Euler Equations
Cauchy-Euler Equation
• A linear differential equation of the form
n
n 1
d
y
d
y
dy
n
n 1
an x
 a n 1 x
 .......  a1 x  a0 y  g ( x)
n
n 1
dx
dx
dx
(1)
where the coefficients an, an-1,….., a2, a1, a0 are constants is known
as a Cauchy –Euler Equations.
• We will look at the general solutions of the homogenous secondorder equations
2
d
y
dy
2
ax
 bx  cy  0
(2)
2
dx
dx
Method of Solution
• We look at the solution of the form xm where we determine m.
• We substitute xm in Equation (2) and each term of the CauchyEuler Equation becomes a polynomial in m times xm .
• Thus, y = xm is a solution of the differential equation whenever
m is a solution of the auxiliary equation.
am(m  1)  bm  c  0
Case 1: Real and Distinct Roots
• Solve
2 x 2 y   3xy   y  0
Case 2: Real and Repeated Roots
• Solve
x 2 y   5xy   4 y  0
Case 3: Conjugate Complex Roots
• Solve
x 2 y   xy   y  0
Example: Higher-Order DE
• Solve
Example: (Use variation of parameters)
• Solve