For what values of $\varepsilon$ has $x^2-1 = \varepsilon e^x$ how many solutions?

Question

Consider the equation $x^2-1 = \varepsilon e^x$. How many solutions does it have dependent of the value of $\varepsilon$? By plotting i guess the number is either 0,1,2 or 3. How does one rigorously show this and how to find the 'turning point' $\tilde \varepsilon$ such that for $\varepsilon > \tilde \varepsilon$ we have 1 solution, for $\varepsilon = \tilde \varepsilon$ we have 2 solutions, for $0 < \varepsilon < \tilde \varepsilon$ we have 3 solutions.

Answer 1

HINT : We have $\frac{x^2-1}{e^x}=\varepsilon $. So, let $f(x)=\frac{x^2-1}{e^x}$ and consider the graph of $y=f(x)$. Then, we have $$f'(x)=\frac{2xe^x-(x^2-1)e^x}{e^{2x}}=\frac{2x-x^2+1}{e^x}.$$ So, we have $f'(x)=0\iff x=1\pm\sqrt 2.$

The answer will be the followings :

• $1$ solution for $\varepsilon \gt f(1+\sqrt 2)$
• $2$ solutions for $\varepsilon =f(1+\sqrt 2)$
• $3$ solutions for $0\lt\varepsilon \lt f(1+\sqrt 2)$
• $2$ solutions for $f(1-\sqrt 2)\lt \varepsilon \le 0$
• $1$ solution for $\varepsilon=f(1-\sqrt 2)$
• $0$ solution for $\varepsilon\lt f(1-\sqrt 2)$

Answer 2

I presume you're looking for real solutions (otherwise there are infinitely many). There are basically two ways the number of real solutions of an equation $f(x,\epsilon) = 0$ can change (where $f$ is real-analytic):

1. Two or more solutions collide.

2. A solution goes off to $+\infty$ or $-\infty$.

   1. When two solutions collide, you have a solution of $f(x, \epsilon) = 0$ coinciding with $\partial f/\partial x = 0$. Then $2x = x^2 - 1$ and $\epsilon = \ldots$.

   2. In this case the only value of $\epsilon$ near which you have to worry about solutions going to $\pm \infty$ is $0$ (do you see why?)

Now see what happens in each of the intervals into which these $\epsilon$ values divide $\mathbb R$.

Answer 3

The two values can be found as

$$ \epsilon_\pm = 2 ( 1 \pm \sqrt{2} ) e^{-( 1 \pm \sqrt{2} )} $$

$$ \epsilon_+ \approx 0.4318431675229668056327158172508330173077... $$

$$ \epsilon_- \approx -1.253559564347305572948775428505549223812... $$