Solution of Constant Coefficient Homogeneous LSOODE/Equal Real Roots of Auxiliary Equation

Theorem
Let $p^2 = 4 q$.

Then $(1)$ has the general solution:


 * $y = C_1 e^{m_1 x} + C_2 x e^{m_1 x}$

Proof
Consider the auxiliary equation of $(1)$:
 * $(2): \quad m^2 + p m + q$

Let $p^2 = 4 q$.

From Solution to Quadratic Equation with Real Coefficients, $(2)$ has one (repeated) root, that is:


 * $m_1 = m_2 = -\dfrac p 2$

From Exponential Function is Solution of Constant Coefficient Homogeneous LSOODE iff Index is Root of Auxiliary Equation:
 * $y_1 = e^{m_1 x}$

is a particular solution to $(1)$.

From Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another:
 * $\map {y_2} x = \map v x \, \map {y_1} x$

where:
 * $\displaystyle v = \int \dfrac 1 { {y_1}^2} e^{-\int P \rd x} \rd x$

is also a particular solution of $(1)$.

We have that:

Hence:

and so:

From Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution:


 * $y = C_1 e^{m_1 x} + C_2 x e^{m_1 x}$