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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:

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.