Finding 5 roots of e^x sin^2 x - cos x = 0 in Mathematica

[vp43]

Here is my problem: I got A right, but B is wrong... Can point out what I'm doing wrong? Thanks!

Develop a Mathematica procedure to find 5 roots of the following equations.

a.) x^5 + 5x^4 + 4x^3 + 3x^2 + 2x + 1 = 0

b.) e^x sin^2 x - cos x = 0

For part A, I got NSolve[x^5 + 5x^4 + 4x^3 + 3x^2 + 2x + 1 == 0, x]

And got 5 roots of the following:
{{x -> -4.19273}, {x -> -0.564099 -
0.390903 \[ImaginaryI]}, {x -> -0.564099 + 0.390903 \[ImaginaryI]}, {x -> \
0.160462\[InvisibleSpace] - 0.693272 \[ImaginaryI]}, {x -> 0.160462\
\[InvisibleSpace] + 0.693272 \[ImaginaryI]}}

But for part B, I did the procedure:

FindRoot[E^x Sin^2 x - Cos x == 0, x]

And gave me: FindRoot::fdss: Search specification x should be a list with a 2-5 elements. (FindRoot[\[ExponentialE]\^x\ Sin\^2\ x - Cos\ x == 0, x]\)

~TRI~

 

[Justin Lazear]

You might also be interested in the TeXForm command if you're going to be posting output from Mathematica much. That way, you only need put in the [ tex ] and [ /tex ] tags.

$$\{ \{ {x\rightarrow {-2.96732}}\} ,<br /> \{ {x\rightarrow {-0.652083 - 0.707484\,\imag }}\} ,<br /> \{ {x\rightarrow {-0.652083 + 0.707484\,\imag }}\} ,<br /> \{ {x\rightarrow {0.135744 - 0.587885\,\imag }}\} ,<br /> \{ {x\rightarrow {0.135744 + 0.587885\,\imag }}\} \}$$

is output from

NSolve[x^5 + 5x^4 + 4x^3 + 3x^2 + 2x + 1 == 0, x] // TeXForm

or

TeXForm[NSolve[x^5 + 5x^4 + 4x^3 + 3x^2 + 2x + 1 == 0, x]]

--J

 

[rayveldkamp]

What you do is replace sin^2(x) by 1-cos^2(x) and you get the DE:
e^x cos^2(x) +cos(x)==e^x,

using the Solve function on Mathematica, BUT don't solve in terms of x solve for Cos[x]
i.e: Solve[e^x cos^2(x) +cos(x)==e^x, Cos[x] ]

You then get solutions in terms of Cos[x], which you can then solve trigonometrically, the reason Mathematica doesn't like the equation is because the there are infintite solutions, and the Solve function can't handle these.

Hope this helps
Ray