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

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Theorem

Let:

$(1): \quad y + p y' + q y = 0$

be a constant coefficient homogeneous linear second order ODE.

Let $m_1$ and $m_2$ be the roots of the auxiliary equation $m^2 + p m + q = 0$.


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:

$\ds v = \int \dfrac 1 { {y_1}^2} e^{-\int P \rd x} \rd x$

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


We have that:

\(\ds \int P \rd x\) \(=\) \(\ds \int p \rd x\)
\(\ds \) \(=\) \(\ds p x\)
\(\ds \leadsto \ \ \) \(\ds e^{-\int P \rd x}\) \(=\) \(\ds e^{-p x}\)
\(\ds \) \(=\) \(\ds e^{2 m_1 x}\)


Hence:

\(\ds v\) \(=\) \(\ds \int \dfrac 1 { {y_1}^2} e^{-\int P \rd x} \rd x\) Definition of $v$
\(\ds \) \(=\) \(\ds \int \dfrac 1 {e^{2 m_1 x} } e^{2 m_1 x} \rd x\) as $y_1 = e^{m_1 x}$
\(\ds \) \(=\) \(\ds \int \rd x\)
\(\ds \) \(=\) \(\ds x\)


and so:

\(\ds y_2\) \(=\) \(\ds v y_1\) Definition of $y_2$
\(\ds \) \(=\) \(\ds x e^{m_1 x}\)


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

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

$\blacksquare$


Sources

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