homogeneous linear differential equation

The linear differential equation

$a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+\mathrm{\dots }+a_1(x)y^{\prime }+a_0(x)y=b(x)$(1)

is calledhomogeneous (http://planetmath.org/HomogeneousLinearDifferentialEquation)iff  $b(x)\equiv 0$.  If  $b(x)\not\equiv 0$,the equation (1) is inhomogeneous.
If (1) is homogeneous (http://planetmath.org/HomogeneousLinearDifferentialEquation),then the sum of any solutions is a solution and any solutionmultiplied by a constant is a solution.

The special case

$$c_n{x}^n{y}^{(n)}+c_{n-1}{x}^{n-1}{y}^{(n-1)}+\mathrm{\dots }+c_{1}x{y}^{\prime }+c_{0}y=\mathrm{\hspace{0.33em}0}$$

of (1), where the $c_i$’s are constants, can via thesubstitution  $x=e^t$  be transformed into a homogeneouslinear differential equation of the same order but withconstant coefficients.