# (D^{3} - 14D + 8) y = 0

$\left({D}^{3}-14D+8\right)y=0$
You can still ask an expert for help

## Want to know more about Laplace transform?

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Cleveland Walters
Step 1. Differential equation
Homogeneous DE with constant coefficient- auxiliary equations:
$\left({D}^{3}-14D+8\right)y=0$
Differential equation:
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Step 2. Solution
$D=m$
${m}^{3}-14m+8=0$
$m=-4$ is a factor,
$⇒{m}^{3}+4{m}^{2}-4{m}^{2}-16m+2m+8=0$
$⇒{m}^{2}\left(n+4\right)-4m\left(m+4\right)+2\left(m+4\right)=0$
$⇒\left(m+4\right)\left({m}^{2}-4m+2\right)=0$
$m=-4$,
$m=\frac{4±\sqrt{16-8}}{2}$
$m=\frac{4±2\sqrt{2}}{2}$
$m=2±\sqrt{2}$
Step 3. Solution
The complementary function is:
${y}_{c}={c}_{1}{e}^{-4x}+{e}^{2x}\left({c}_{2}\mathrm{cos}\sqrt{2x}+{c}_{3}\mathrm{sin}\sqrt{2x}\right)$
${c}_{1},{c}_{2},{c}_{3}$ are arbitrary constant.
###### Not exactly what you’re looking for?
Mason Hall
Given that,
$\left({D}^{2}-14D+8\right)y=0$
The auxilary equation of above eq-n is-
${m}^{3}-14m+8=0$
$⇒{m}^{3}+4m-4{m}^{2}-16m+2m+g=0$
$⇒{m}^{2}\left(m+4\right)-4m\left(m+4\right)+2\left(m+4\right)=0$
$⇒\left(m+4\right)\left({m}^{2}-4m=2\right)=0$
$⇒\left(m+4\right)\left({m}^{2}-4m+2\right)=0$

so, the roots of auxilary eq-n are real and distinct
###### Not exactly what you’re looking for?
RizerMix

$d=-4$
$d=\frac{4-\sqrt{8}}{2}=2-\sqrt{2}=0.586$
$d=\frac{4+\sqrt{8}}{2}=2+\sqrt{2}=3.414$
$y=0$