Slide 1
Cauchy-Euler
Equations
Differential Equations
Section 4.7
Slide 2
Standard Form of a CauchyEuler Equation
an x n y ( n )  an 1 x n 1 y ( n 1)    a2 x 2 y  a1 xy  a0 y  g ( x)
We will try to find a solution of the form
y  xm
Slide 3
Case I

m1 and m2 are distinct real roots

The solution has the form
y  c1 x m1  c2 x m2

Example: #6
Slide 4
Case II

m1 and m2 are repeated real roots

The solution looks like
y  c1 x m1  c2 x m1 ln x

Example: #2
Slide 5
Case III

m1 and m2 are conjugate complex roots

The solution looks like
y  x c1 cos(  ln x)  c2 sin(  ln x)

Example: # 14
Slide 6
Higher Order Equations

Extend the form of the solution as we
did for equations with constant
coefficients. Refer to the top of page
196 if necessary.