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

Problem Statement

Find a general solution. Check your answer by substitution.

Problem 8

$y^{″}+y^{\prime }+3.25y=0$

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

$y^{\prime }(x)=-\frac{1}{2}{e}^{-\frac{1}{2}x}(c_1\cos (x\sqrt{3})+c_2\sin (x\sqrt{3})+e^{-\frac{1}{2}x}(-\sqrt{3}{c}_1\sin \sqrt{3}x+\sqrt{3}{c}_1\cos \sqrt{3}x)$
$y^{″}(x)=\frac{1}{4}{e}^{-\frac{1}{2}x}(c_1\cos (x\sqrt{3})+c_2\sin (x\sqrt{3})-\frac{1}{2}{e}^{-\frac{1}{2}x}(-\sqrt{3}{c}_1\sin \sqrt{3}x+\sqrt{3}{c}_1\cos \sqrt{3}x)-$
$\frac{1}{2}{e}^{-\frac{1}{2}x}(-\sqrt{3}{c}_1\sin \sqrt{3}x+\sqrt{3}{c}_1\cos \sqrt{3}x)e^{-\frac{1}{2}x}(-3c_1\cos (x\sqrt{3})-3c_2\sin (x\sqrt{3})$
Substituting $y,y^{\prime },y^{″}$ into the original equation, the result is

 $y^{″}+y^{\prime }+3.25y=0$

Problem 15

$y^{″}+0.54y^{\prime }+(0.0729+\pi )y=0$
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

$y^{\prime }(x)=-0.27e^{-0.27x}(c_1\cos (x\sqrt{\pi })+c_2\sin (x\sqrt{\pi })+e^{-0.27x}(-\sqrt{\pi }{c}_1\sin \sqrt{\pi }x+\sqrt{\pi }{c}_1\cos \sqrt{\pi }x)$

$y^{″}(x)=0.0729e^{-0.27x}(c_1\cos (x\sqrt{\pi })+c_2\sin (x\sqrt{\pi })-0.27e^{-0.27x}(-\sqrt{\pi }{c}_1\sin \sqrt{\pi }x+\sqrt{\pi }{c}_1\cos \sqrt{\pi }x)-$

$0.27e^{-0.27x}(\sqrt{\pi }{c}_1\sin \sqrt{\pi }x+\sqrt{\pi }{c}_1\cos \sqrt{\pi }x)+e^{-0.27x}(-\pi (c_1\cos (x\sqrt{\pi }))-\pi (c_2\sin (x\sqrt{\pi })))$

Substituting $y,y^{\prime },y^{″}$ into the original equation, the result is

 $y^{″}+0.54y^{\prime }+(0.0729+\pi )y=0$

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