Linear Second Order ODE/y'' - 7 y' - 5 y = 0

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Theorem

The second order ODE:

$(1): \quad y'' - 7 y' - 5 y = 0$

has the general solution:

$y = C_1 \, \map \exp {\paren {\dfrac 7 2 + \dfrac {\sqrt {69} } 2} x} + C_2 \, \map \exp {\paren {\dfrac 7 2 - \dfrac {\sqrt {69} } 2} x}$


Proof

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

Its auxiliary equation is:

$(2): \quad: m^2 - 7 m - 5 = 0$

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

\(\ds m\) \(=\) \(\ds \dfrac {7 \pm \sqrt {7^2 - 4 \times 1 \times \paren {-5} } } 2\)
\(\ds \) \(=\) \(\ds \dfrac 7 2 \pm \dfrac {\sqrt {69} } 2\)


These are real and unequal.

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

$y = C_1 \, \map \exp {\paren {\dfrac 7 2 + \dfrac {\sqrt {69} } 2} x} + C_2 \, \map \exp {\paren {\dfrac 7 2 - \dfrac {\sqrt {69} } 2} x}$

$\blacksquare$