Linear Second Order ODE/y'' + 2 b y' + a^2 y = 0/b less than a

From ProofWiki
Jump to navigation Jump to search

Theorem

The second order ODE:

$(1): \quad y'' + 2 b y' + a^2 y = 0$ where $b^2 < a^2$

has the general solution:

$y = e^{-b x} \paren {C_1 \, \map \cos {\sqrt {a^2 - b^2} } x + C_2 \, \map \sin {\sqrt {a^2 - b^2} } x}$


Proof

It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.

Its auxiliary equation is:

$(2): \quad: m^2 + 2 b m + a^2 = 0$

From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:

$m_1 = - b + \sqrt {b^2 - a^2}$
$m_2 = - b - \sqrt {b^2 - a^2}$

As $b^2 < a^2$, this converts into:

$m_1 = - b + i \sqrt {a^2 - b^2}$
$m_2 = - b - i \sqrt {a^2 - b^2}$


These are complex and unequal.

So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:

$y = e^{-b x} \paren {C_1 \, \map \cos {\sqrt {a^2 - b^2} } x + C_2 \, \map \sin {\sqrt {a^2 - b^2} } x}$

$\blacksquare$