Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE

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Theorem

Let $c_1$ and $c_2$ be real numbers.

Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE:

$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$


Then:

$c_1 \, \map {y_1} x + c_2 \, \map {y_2} x$

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


That is, a linear combination of particular solutions to a homogeneous linear second order ODE is also a particular solution to that ODE.


Proof

\(\ds \) \(\) \(\ds \paren {c_1 \, \map {y_1} x + c_2 \, \map {y_2} x}' ' + \map P x \paren {c_1 \, \map {y_1} x + c_2 \, \map {y_2} x}' + \map Q x \paren {c_1 \, \map {y_1} x + c_2 \, \map {y_2} x}\)
\(\ds \) \(=\) \(\ds \paren {c_1 \, \map {y_1' '} x + c_2 \, \map {y_2' '} x} + \map P x \paren {c_1 \, \map {y_1'} x + c_2 \, \map {y_2'} x} + \map Q x \paren {c_1 \, \map {y_1} x + c_2 \, \map {y_2} x}\) Linear Combination of Derivatives
\(\ds \) \(=\) \(\ds c_1 \paren {\map {y_1' '} x + \map P x \, \map {y_1'} x + \map Q x \, \map {y_1} x} + c_2 \paren {\map {y_2' '} x + \map P x \, \map {y_2'} x + \map Q x \, \map {y_2} x}\)
\(\ds \) \(=\) \(\ds c_1 \cdot 0 + c_2 \cdot 0\)
\(\ds \) \(=\) \(\ds 0\)

Hence the result.

$\blacksquare$


Sources

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