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* = *x *^{m} and solve for *m*. If *y* = *x *^{m} , 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* = *e *^{mx} 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* = *x *^{m} 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 *m* _{1} and *m* _{2}, 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* = *x *^{m} results in

Since the roots of the resulting quadratic equation are real and distinct (Case 1), both *y* = *x* ^{1} = *x* and *y* = *x* ^{3} 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* = *x *^{m}

Since the roots of the resulting quadratic equation are real and identical (Case 2), both *y* = *x* ^{2} and *y* = *x* ^{2} 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 *non*homogeneous equidimensional equation is desired, first use the method above to obtain the general solution of the corresponding homogeneous equation; then apply variation of parameters.