Solve $9\cosh(x) - 5\sinh(x) = 9$

Question

I am having trouble solving $$9\cosh(x)-5\sinh(x)=9$$

I have done these steps already: $$9\left(\frac{e^x+e^{-x}}{2}\right) - 5\left(\frac{e^x-e^{-x}}{2}\right)=9$$ $$\frac{9e^x +9e^{-x}-5e^x+5e^{-x}}{2} =9 $$ $$2e^x + 7e^{-x} = 9$$

I am stuck on where to go next. (Provided the steps i've taken before are correct)

Thanks

Let $y=e^{x}$ and solve for $y$ first. – Thomas Andrews Mar 22 '17 at 19:57 — Thomas Andrews, Mar 22 '17 at 19:57

score 1 · Accepted Answer · answered Mar 22 '17 at 19:58

1

HINT: multiply your equation by $$e^x$$ and solve a quadratic equation the equation is given by $$2e^{2x}-9e^x+7=0$$ can you finish this?

answered Mar 22 '17 at 19:58

Dr. Sonnhard Graubner

95,283

So simple yet I couldn't see it. Thank you – MathsRookie Mar 22 '17 at 19:59
you are welcome, some simple Things we can not see in the world – Dr. Sonnhard Graubner Mar 22 '17 at 20:00

SchrodingersCat · Answer 2 · 2017-03-23T10:43:49.830

1

Take $e^{x}=p$.

Then your equation becomes $$2p+\frac{7}{p}=9$$ $$2p^2-9p+7=0$$ $$p=\frac{9\pm\sqrt{81-56}}{4}=\frac{9\pm5}{4}=1,\frac{14}{4}$$ $$x=\ln \left(\frac{9\pm5}{4}\right)=0,\ln \frac{14}{4}$$

Hope this helps you.

edited Mar 23 '17 at 10:43

answered Mar 22 '17 at 20:02

SchrodingersCat

24,472

Funny, I thought $p=1$ ($x=0$) was a solution... – Did Mar 23 '17 at 06:08

score 0 · Answer 3 · answered Mar 22 '17 at 19:58

0

Hint : multiply by $y=e^x$ and solve the equation with $y$

answered Mar 22 '17 at 19:58

Vincent

2,318

score 0 · Answer 4 · answered Mar 22 '17 at 19:59

0

Hint: Your equation is: $$ \frac{2e^{2x}-9e^x+7}{e^x}=0 $$ solve for $e^x=y$

answered Mar 22 '17 at 19:59

Emilio Novati

62,675

Solve $9\cosh(x) - 5\sinh(x) = 9$

4 Answers4