An order linear ordinary differential equation with variable coefficients has the general form of
Most ordinary differential equations with variable coefficients are not possible to solve by hand. However, some special cases do exist:
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The Euler-Cauchy differential equation has the general form of
where are constants and the power of is always equal to the order of the derivative of in each term.
To solve this problem, let , the derivatives of become
The Euler-Cauchy differential equation can therefore be simplified to a linear homogeneous or non-homogeneous ODE with constant coefficients.
At the end, the variable must be changed back to .
Consider an order linear ordinary differential equation with variable coefficients with the general form of
If
the above differential equation is an order linear exact differential equation which can be rewritten as
The order of this differential equation can hence be reduced by direct integration.
The method of variation of parameters can be used to obtain the particular solution when the complementary solution is known. Refer to the section of the Method of Variation of Parameters for further detail.
are constants and the power of 
is always equal to the order of the derivative of 
in each term.
, the derivatives of 
must be changed back to 
