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