Second Order Homogeneous Equations with Repeated Roots

Derive the general form solution for a second-order constant coefficient linear homogeneous differential equation with repeated roots.

In solving second-order constant coefficient linear homogeneous differential equations, one encounters a key concept when the characteristic equation has repeated roots. This situation requires a distinct approach compared to equations with distinct roots. When the characteristic polynomial of the differential equation has repeated roots, say r1, a straightforward exponential solution fails to account for the multiplicity of the root. Instead, the solution involves not only the exponential function associated with the root but also another term that involves multiplication by the independent variable, often denoted as x. This leads to a general solution structure which caters to repeated roots effectively.

The rationale behind this formulation lies in the nature of linear dependence among solutions of homogeneous differential equations. Given that the solutions must form a linearly independent set, the term involving x helps create the necessary independence from the exponential solution. This allows us to span the solution space appropriately, ensuring that the general solution encompasses all possible solutions to the equation. Understanding this concept is crucial, as it underpins more complex systems and forms the basis for addressing non-homogeneous equations with repeated roots as well. This involves not only mastering the mechanics of the solution process but also appreciating the underlying algebraic principles.