The second-order homogeneous
Cauchy-Euler equidimensional
equation has the form
where
a, b, and
c are constants (and
a ≠ 0). The quickest way to solve this linear equation is to is to substitute
y =
xm
and solve for
m. If
y =
xm
, then
so substitution into the differential equation yields
Just as in the case of solving second-order linear homogeneous equations with constant coefficients (by first setting
y =
emx
and then solving the resulting auxiliary quadratic equation for
m), this process of solving the equidimensional equation also yields an auxiliary quadratic polynomial equation. The question here is, how is
y =
xm
to be interpreted to give two linearly independent solutions (and thus the general solution) in each of the three cases for the roots of the resulting quadratic equation?
Case 1: The roots of (*)
are real and distinct.
If the two roots are denoted
m1 and
m2, then the general solution of the second-order homogeneous equidimensional differ-ential equation in this case is
Case 2: The roots of (*)
are real and identical.
If the double (repeated) root is denoted simply by
m, then the general solution (for
x > 0) of the homogeneous equidimensional differential equation in this case is
Case 3: The roots of (*)
are distinct conjugate complex numbers.
If the roots are denoted
r ±
si, then the general solution of the homogeneous equidimensional differential equation in this case is
Example 1: Give the general solution of the equidimensional equation
Substitution of
y =
xm
results in
Since the roots of the resulting quadratic equation are real and distinct (Case 1), both
y =
x1 =
x and
y =
x3 are solutions and linearly independent, and the general solution of this homogeneous equation is
Example 2: For the following equidimensional equation, give the general solution which is valid in the domain
x > 0:
Substitution of
y =
xm
Since the roots of the resulting quadratic equation are real and identical (Case 2), both
y =
x2 and
y =
x2 In
x are (linearly independent) solutions, so the general solution (valid for
x > 0) of this homogeneous equation is
If the general solution of a
nonhomogeneous equidimensional equation is desired, first use the method above to obtain the general solution of the corresponding homogeneous equation; then apply variation of parameters.
Review and Introduction
Second-Order Equations
