Linear Second Order ODE/y'' + 8 y = 0
Jump to navigation
Jump to search
Theorem
The second order ODE:
- $(1): \quad y + 8 y = 0$
has the general solution:
- $y = C_1 \cos 2 \sqrt 2 x + C_2 \sin 2 \sqrt 2 x$
Proof
It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
- $(2): \quad: m^2 + 8 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
- $m_1 = 2 \sqrt 2 i$
- $m_2 = -2 \sqrt 2 i$
These are complex and unequal.
So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
- $y = C_1 \cos 2 \sqrt 2 x + C_2 \sin 2 \sqrt 2 x$
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.17$: Problem $1 \ \text{(c)}$