Constant Coefficient Homogeneous Differential Equations Repeated Roots

We discuss homogeneous second order constant coefficient ODEs. We introduce the differential operator, D, and explain how the polynomial differential operator, p(D), can be factored and how its factors commute. This method allows us to find the general solution to the homogeneous second order constant coefficient ODE.

This is referred to as the method of differential annihilation operators.

We consider the case when the polynomial has repeated real roots. This is the case when the characteristic polynomial has real repeated roots.

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[JMCoster]

Now I understand where the x in c1xe^2x is coming from. Thanks !