In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential equation can be solved explicitly. The equation Let be the nth derivative of the unknown function . Then a Cauchy–Euler equation of order n has the form The substitution (that is, ; for , one might replace all instances of by , which extends the solution's domain to ) may be used to reduce this equation to a linear differential equation with constant coefficients. Alternatively, the trial solution may be used to directly solve for the basic solutions. Second order – solving through trial solution The most common Cauchy–Euler equation is the second-order equation, appearing in a number of physics and engineering applications, such as when solving Laplace's equation in polar coordinates. The second order Cauchy–Euler equation is We assume a trial solution Differentiating gives and Substituting into the original equation leads to requiring Rearranging and factoring gives the indicial equation We then solve for m. There are three particular cases of interest: Case 1 of two distinct real roots, and ; Case 2 of one real repeated root, ; Case 3 of complex roots, . In case 1, the solution is In case 2, the solution is To get to this solution, the method of reduction of order must be applied after having found one solution . In case 3, the solution is For . This form of the solution is derived by setting and using Euler's formula Second order – solution through change of variables We operate the variable substitution defined by Differentiating gives Substituting the differential equation becomes This equation in is solved via its characteristic polynomial Now let and denote the two roots of this polynomial. We analyze the case where there are distinct roots and the case where there is a repeated root: If the roots are distinct, the general solution is where the exponentials may be complex. If the roots are equal, the general solution is In both cases, the solution may be found by setting . Hence, in the first case, and in the second case, Example Given (2x+1)²y'' - 6(2x+1)y' + 16y =8(2x+1)² we substitute the simple solution : For to be a solution, either , which gives the trivial solution, or