University of Florida/Egm4313/s12.team11.gooding/R2/2.8

Problem R2.8

Problem Statement

Find a general solution. Check your answer by substitution.

Problem 8

$y''+y'+3.25y=0\!$

Let:
$\lambda =d/dx\!$

Characteristic Equation

$\lambda ^{2}+\lambda +3.25=0\!$
Using the quadratic equation to find roots we get:
$\lambda _{1}={\frac {-1+i{\sqrt {(}}12)}{2}}\!$
$\lambda _{2}={\frac {-1-i{\sqrt {(}}12)}{2}}\!$
Therefore:

$y_{h}(x)=e^{-{\frac {1}{2}}x}(c_{1}\cos(x{\sqrt {3}})+c_{2}\sin(x{\sqrt {3}})\!$

Check By Substitution

$y'(x)=-{\frac {1}{2}}e^{-{\frac {1}{2}}x}(c_{1}\cos(x{\sqrt {3}})+c_{2}\sin(x{\sqrt {3}})+e^{-{\frac {1}{2}}x}(-{\sqrt {3}}c_{1}\sin {{\sqrt {3}}x}+{\sqrt {3}}c_{1}\cos {{\sqrt {3}}x})\!$
$y''(x)={\frac {1}{4}}e^{-{\frac {1}{2}}x}(c_{1}\cos(x{\sqrt {3}})+c_{2}\sin(x{\sqrt {3}})-{\frac {1}{2}}e^{-{\frac {1}{2}}x}(-{\sqrt {3}}c_{1}\sin {{\sqrt {3}}x}+{\sqrt {3}}c_{1}\cos {{\sqrt {3}}x})-\!$
${\frac {1}{2}}e^{-{\frac {1}{2}}x}(-{\sqrt {3}}c_{1}\sin {{\sqrt {3}}x}+{\sqrt {3}}c_{1}\cos {{\sqrt {3}}x})e^{-{\frac {1}{2}}x}(-3c_{1}\cos(x{\sqrt {3}})-3c_{2}\sin(x{\sqrt {3}})\!$
Substituting $y,y',y''\!$ into the original equation, the result is

 $y''+y'+3.25y=0\!$

Problem 15

$y''+0.54y'+(0.0729+\pi )y=0\!$
Let: ${\frac {d}{dx}}=\lambda \!$

Characteristic Equation

$\lambda ^{2}+0.54\lambda +(0.0729+\pi )=0\!$
Using the quadratic equation to find roots we get:
$\lambda _{1}={\frac {-0.27+i{\sqrt {(}}\pi )}{2}}\!$
$\lambda _{2}={\frac {-0.27-i{\sqrt {(}}\pi )}{2}}\!$
Therefore:

$y_{h}(x)=e^{-0.27x}(c_{1}\cos(x{\sqrt {\pi }})+c_{2}\sin(x{\sqrt {\pi }})\!$

Check By Substitution

$y'(x)=-0.27e^{-0.27x}(c_{1}\cos(x{\sqrt {\pi }})+c_{2}\sin(x{\sqrt {\pi }})+e^{-0.27x}(-{\sqrt {\pi }}c_{1}\sin {{\sqrt {\pi }}x}+{\sqrt {\pi }}c_{1}\cos {{\sqrt {\pi }}x})\!$

$y''(x)=0.0729e^{-0.27x}(c_{1}\cos(x{\sqrt {\pi }})+c_{2}\sin(x{\sqrt {\pi }})-0.27e^{-0.27x}(-{\sqrt {\pi }}c_{1}\sin {{\sqrt {\pi }}x}+{\sqrt {\pi }}c_{1}\cos {{\sqrt {\pi }}x})-\!$

$0.27e^{-0.27x}({\sqrt {\pi }}c_{1}\sin {{\sqrt {\pi }}x}+{\sqrt {\pi }}c_{1}\cos {{\sqrt {\pi }}x})+e^{-0.27x}(-\pi (c_{1}\cos(x{\sqrt {\pi }}))-\pi (c_{2}\sin(x{\sqrt {\pi }})))\!$

Substituting $y,y',y''\!$ into the original equation, the result is

 $y''+0.54y'+(0.0729+\pi )y=0\!$

Egm4313.s12.team11.gooding 03:41, 7 February 2012 (UTC)