(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider [itex] x = Ae^{at} + Be^{bt} [/itex]

a) Show this can have at most one zero.

b) Show that if x(t)=0 for t>0, then either x(0)>0 and x'(0)<0; or x(0)<0 and x'(0)>0

3. The attempt at a solution

a)

x = Ae^{at} + Be^{bt}

0 = Ae^{at} + Be^{bt}

0 = ln(Ae^{at} + Be^{bt}}

0 = ln(A) + ln(e^{at}) + ln(B) + e^{bt}

0 = ln(A) + ln(B) + at + bt

0 = ln(AB) + t(a+b)

t=\frac{-ln(AB)}{a+b}

A,B,a, and b are constants so that's a distinct solution and the equation has just one zero, right?

b)

t>0

a,b<0 (I think that was given in class)

x = Ae^{at} + Be^{bt} [/itex] [itex]

x' = aAe^{at} + bBe^{bt}

so after some manipulation I got

x = ln(AB) + t(a+b)

x' = ln(abAB) + t(a+b)

In both these equations, I think the first term would be positive and the second term negative. I don't see how to reach the "either x(0)>0 and x'(0)<0; or x(0)<0 and x'(0)>0" conclusion. Any help?

Thank you very much.

p.s. I was trying to use latex for this post but couldn't get line breaks to work. How do I get equations on different lines? I tried \\ but nothing happened.

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# Homework Help: Constant coefficient equation

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