How should I show that this root is given approximately by this?

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Homework Statement
By sketching the graphs of ## y=a^2-x^2 ## and ## y=\epsilon\sinh x ## for ## x\geq 0 ##, show that the equation ## a^2-x^2=\epsilon\sinh x, 0<\epsilon<<1 ##, has a root near ## x=a ##. Use perturbation theory to show that this root is given approximately by ## x=a-\frac{\epsilon}{2a}\sinh a+\frac{\epsilon^{2}}{16a^3}(2a\sinh 2a-\cosh 2a+1)+O(\epsilon^{3}) ##.
Relevant Equations
None.
Proof:

Consider the equation of ## a^2-x^2=\epsilon\sinh x ## for ## 0<\epsilon<<1 ##.
Let ## \epsilon=0 ##.
Then the unperturbed equation is ## a^2-x^2=0 ##.
This gives ## a^2-x^2=0\implies (a+x)(a-x)=0\implies x=\pm a ## with the root ## x=x_{0} ##
such that ## x_{0}(a)=a ## because ## x\geq 0 ## implies that ## x\neq -a ##.
Applying the perturbation theory produces ## x=a+\epsilon x_{1}+\epsilon^{2}x_{2}+O(\epsilon^{3}) ##
for ## 0<\epsilon<<1 ##.
By direct substitution of this expansion ## x=a+\epsilon x_{1}+\epsilon^{2}x_{2}+O(\epsilon^{3}) ##
into the equation ## a^2-x^2=\epsilon\sinh x, 0<\epsilon<<1 ## for ## x\geq 0 ##,
we get ## a^2-(a+\epsilon x_{1}+\epsilon^{2}x_{2}+O(\epsilon^{3}))^{2}=
\epsilon\sinh(a+\epsilon x_{1}+\epsilon^{2}x_{2}+O(\epsilon^{3})) ## for ## 0<\epsilon<<1 ##
where ## x\geq 0 ##.
Observe that ## a^2-(a+\epsilon x_{1}+\epsilon^{2}x_{2}+O(\epsilon^{3}))^{2}=
a^2-(a^2+a\epsilon x_{1}+a\epsilon^{2}x_{2}+aO(\epsilon^{3})+a\epsilon x_{1}+\epsilon^{2}x_{1}^{2}
+\epsilon^{3}x_{1}x_{2}+\epsilon x_{1}O(\epsilon^{3})+a\epsilon^{2}x_{2}
+\epsilon^{3}x_{1}x_{2}+
\epsilon^{4}x_{2}^{2}+\epsilon^{2}x_{2}O(\epsilon^{3})+aO(\epsilon^{3})+\epsilon x_{1}O(\epsilon^{3})
+\epsilon^{2}x_{2}O(\epsilon^{3})+(O(\epsilon^{3}))^{2} ##.
Thus, ## -2a\epsilon x_{1}-2a\epsilon^{2}x_{2}
-\epsilon^{2}x_{1}^{2}-2\epsilon^{3}x_{1}x_{2}-\epsilon^{4}x_{2}^{2}+O(\epsilon^{3}) ##.
Similarly, we will use the Taylor's series to expand ## \sinh(a+\epsilon x_{1}+\epsilon^{2}x_{2}+O(\epsilon^{3})) ##
where the Taylor's series expansion for the hyperbolic function is given by
## \sinh(x)=x+\frac{x^3}{3!}+\frac{x^5}{5!}+... ##.

For the first part of the question/problem, I've already obtained the sketch of the graphs of those two given
equations from my graphing calculator but I have doubt about the second part of the question/problem
where the proof is involved because I need to find both ## x_{1}, x_{2} ## in order to show the given
approximation but I want to know, how to use this direct substitution by using the Taylor's series expansion
of ## \sinh(x)=x+\frac{x^3}{3!}+\frac{x^5}{5!}+... ##? Or is this method even correct? May anyone please
check/verify/confirm to see if the work/proof shown above is correct?
 
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Expansion of x by ##\epsilon##
x=a+\epsilon x^{(1)}+\epsilon^2x^{(2)}+...
\sinh x= \sinh a + \epsilon x^{(1)} \cosh a + ...
by Taylor series. The equation
x^2-a^2=-\epsilon \sinh x
(\epsilon x^{(1)}+\epsilon^2 x^{(2)}+...)(2a+\epsilon x^{(1)}+\epsilon^2 x^{(2)}+...)
=-\epsilon\sinh a - \epsilon^2 x^{(1)} \cosh a + ...
Comparing the both sides in order ##\epsilon## terms
2ax^{(1)}=-\sinh a
in order ##\epsilon^2## terms
2ax^{(2)}+(x^{(1)})^2=- x^{(1)} \cosh a
 
Last edited:
anuttarasammyak said:
Expansion of x by ##\epsilon##
x=a+\epsilon x^{(1)}+\epsilon^2x^{(2)}+...
\sinh x= \sinh a + \epsilon x^{(1)} \cosh a + ...
by Taylor series. The equation
x^2-a^2=-\epsilon \sinh x
(\epsilon x^{(1)}+\epsilon^2 x^{(2)}+...)(2a+\epsilon x^{(1)}+\epsilon^2 x^{(2)}+...)
=-\epsilon\sinh a - \epsilon^2 x^{(1)} \cosh a + ...
Comparing the both sides in order ##\epsilon## terms
2ax^{(1)}=-\sinh a
in order ##\epsilon^2## terms
2ax^{(2)}+(x^{(1)})^2=- x^{(1)} \cosh a
I finally solved this problem!
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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