Math 260

1
Math 3120 Differential Equations withBoundary
Value Problems
Chapter 4 Higher-Order Differential
Equations Section 4-7 Cauchy-Euler Equations
2
Cauchy-Euler Equation

• A linear differential equation of the form
• 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 second-order equations

3
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 Cauchy-Euler 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.

4
Case 1 Real and Distinct Roots

• Solve

5
Case 2 Real and Repeated Roots

• Solve

6
Case 3 Conjugate Complex Roots

• Solve

7
Example Higher-Order DE

• Solve

8
Example (Use variation of parameters)

• Solve