# homogeneous linear differential equation

 $\displaystyle a_{n}(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+\ldots+a_{1}(x)y^{\prime}+a_% {0}(x)y\;=\;b(x)$ (1)

is called (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 solution multiplied by a constant is a solution.

The special case

 $c_{n}x^{n}y^{(n)}+c_{n-1}x^{n-1}y^{(n-1)}+\ldots+c_{1}xy^{\prime}+c_{0}y\;=\;0$

of (1), where the $c_{i}$’s are constants, can via the substitution  $x=e^{t}$  be transformed into a homogeneous linear differential equation of the same order but with constant coefficients.

Title homogeneous linear differential equation HomogeneousLinearDifferentialEquation 2014-02-27 10:07:04 2014-02-27 10:07:04 pahio (2872) pahio (2872) 3 pahio (2872) Definition msc 34A05