Zerrilloh6
2021-12-16
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Cleveland Walters

Answered 2021-12-17
Author has **40** answers

Step 1. Differential equation

Homogeneous DE with constant coefficient- auxiliary equations:

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

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,

$\Rightarrow {m}^{3}+4{m}^{2}-4{m}^{2}-16m+2m+8=0$

$\Rightarrow {m}^{2}(n+4)-4m(m+4)+2(m+4)=0$

$\Rightarrow (m+4)({m}^{2}-4m+2)=0$

$m=-4$ ,

$m=\frac{4\pm \sqrt{16-8}}{2}$

$m=\frac{4\pm 2\sqrt{2}}{2}$

$m=2\pm \sqrt{2}$

Step 3. Solution

The complementary function is:

${y}_{c}={c}_{1}{e}^{-4x}+{e}^{2x}({c}_{2}\mathrm{cos}\sqrt{2x}+{c}_{3}\mathrm{sin}\sqrt{2x})$

$c}_{1},{c}_{2},{c}_{3$ are arbitrary constant.

Homogeneous DE with constant coefficient- auxiliary equations:

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

Step 3. Solution

The complementary function is:

Mason Hall

Answered 2021-12-18
Author has **36** answers

Given that,

$({D}^{2}-14D+8)y=0$

The auxilary equation of above eq-n is-

${m}^{3}-14m+8=0$

$\Rightarrow {m}^{3}+4m-4{m}^{2}-16m+2m+g=0$

$\Rightarrow {m}^{2}(m+4)-4m(m+4)+2(m+4)=0$

$\Rightarrow (m+4)({m}^{2}-4m=2)=0$

$\Rightarrow (m+4)({m}^{2}-4m+2)=0$

$\Rightarrow m+4=0\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{m}^{2}-4m+2=0$

$\Rightarrow m=-4\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}m=3.414,m=0.586$

so, the roots of auxilary eq-n are real and distinct

The auxilary equation of above eq-n is-

so, the roots of auxilary eq-n are real and distinct

RizerMix

Answered 2021-12-29
Author has **438** answers

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