- #1

- 89

- 1

y” + (K/m)y = (Kl[tex]^{0}[/tex]+mg)/m

y(0) = l[tex]_{0}[/tex]

y(t[tex]_{e}[/tex]) = (K l[tex]_{0}[/tex]+mg)/K

## The Attempt at a Solution

y(t) = -(mg/K)cos{[tex]\sqrt{K/m}[/tex] t} + (mg/K){cos{[tex]\sqrt{K/m}[/tex] t[tex]_{e}[/tex]}/sin{[tex]\sqrt{K/m}[/tex] t[tex]_{e}[/tex]}}*sin{[tex]\sqrt{K/m}[/tex] t} + (K l[tex]_{0}[/tex]+mg)/K

which works. But the auxiliary equation has imaginary roots and one of the texts I'm reading states that

y(t) = (C[tex]_{1}[/tex] + C[tex]_{2}[/tex])cos{[tex]\sqrt{K/m}[/tex] t} + i(C[tex]_{1}[/tex] - C[tex]_{2}[/tex])sin{[tex]\sqrt{K/m}[/tex] t} + (K l[tex]_{0}[/tex]+mg)/K

should be a solution w/imaginary coefficients. I get

C[tex]_{1}[/tex] = -mg/K - C[tex]_{2}[/tex]

C[tex]_{2}[/tex] = (-i/2)(mg/K)(cos{[tex]\sqrt{K/m}[/tex] t[tex]_{e}[/tex]}/sin{[tex]\sqrt{K/m}[/tex] t[tex]_{e}[/tex]}

but they don't solve the original condition reqs. What am I doing wrong/not seeing?

P.S. Why don't some of the Latex scripts work from the pull down menu; ie subscript gives superscript?

I posted this earlier, but cannot retrieve the posting. I'm still having trouble with the Latex formatting. Subscript gives superscript for some reason. And I'm using the full down menu option and inserting the arg. into the script. (?) Also, regarding the coefficients, please see the following, Case III:

http://www.stewartcalculus.com/data...ntexts/upfiles/3c3-2ndOrderLinearEqns_Stu.pdf