Answer to Question #198204 in Calculus for desmond

Question #198204

2. (a) Express 5 sinh x + cosh x in the form Ae^x+Be^{-x}, where A and B are integers.

(b) Solve the equation 5 sinh x + cosh x + 5 = 0, giving your answer in the form ln a, where a ∈ R.


1
Expert's answer
2021-05-25T17:27:50-0400

(a)  5 sinh x + cosh x

=5.exex2+ex+ex2=5.\frac{e^x-e^{-x}}{2}+\frac{e^x+e^{-x}}{2}

=52ex52ex+12ex+12ex=62ex+42ex=\frac{5}{2}e^x-\frac{5}{2}e^{-x}+\frac{1}{2}e^x+\frac{1}{2}e^{-x} \\ =\frac{6}{2}e^x+\frac{-4}{2}e^{-x}

A=3,B=2\boxed{A=3 , B=-2}


(b) 5 sinh x + cosh x + 5 = 0

=5.exex2+ex+ex2+5=0=5.\frac{e^x-e^{-x}}{2}+\frac{e^x+e^{-x}}{2}+5=0

let exe^x = t

than

52(t1t)+12(t+1t)+5=0\frac{5}{2}(t-\frac{1}{t})+\frac{1}{2}(t+\frac{1}{t})+5=0

    3t2+5t2=0\implies3t^2+5t-2=0

t=5±496t=\frac{-5\pm\sqrt{49}}{6}

ex=5+76e^x=\frac{-5+7}{6}\becauseex>0e^x>0 (always)

x=ln(13)\boxed{x=ln(\frac{1}{3})}

here a=13\frac{1}{3}


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