4
$\begingroup$

I'm asking this as an absolute beginner (second year undergrad student). Suppose I write an equation out of the blue such as $$\log(\tan(\sin x + 23e^{2x+83} +\tan x))= 1000$$, which is non-linear, how would a mathematician attempt to solve it? I am encountering subjects which get more and more abstract(linear algebra, group theory) and I wonder if they are applicable to such problems.

$\endgroup$
8
  • 9
    $\begingroup$ I doubt any mathematician knows how to solve this equation. $\endgroup$
    – Mark
    Commented May 24, 2023 at 9:42
  • 1
    $\begingroup$ We can technically make it $\sin(x) + 23e^{2x+83} +\tan(x)=\arctan(10^{1000})$, but that sounds kind of absurd 🙂 $\endgroup$ Commented May 24, 2023 at 9:43
  • 4
    $\begingroup$ You can use methods such as Newton's method to compute an approximate solution. $\endgroup$
    – littleO
    Commented May 24, 2023 at 9:44
  • 2
    $\begingroup$ In general, a random equation you write up might not have a "pretty" solution, however, it is possible to get an approximate solution with arbitrary precision. Namely you can use Newton's method: Let $f(x)$ be the left side of the equation. Then you can solve $f(x)=0$ by taking an initial value $x_0$, and iterating $x_{k+1} \leftarrow x_k-\frac{f(x_k)}{f'(x_k)}$, where $f'$ is the derivative of the function. The more iterations, the better the precision. To get other solutions, you can try starting from different initial $x_0$ values. $\endgroup$
    – Daniel P
    Commented May 24, 2023 at 9:55
  • 2
    $\begingroup$ An alternative to Newton's method (if let's say, $f$ is non-differentiable), is the slower, Binary search method. Let's say $f(a)>0$ and $f(b)<0$. Then if $f$ is continuous, then there must be a value between $a$ and $b$ (call it $c$), where $f(c)=0$. You can keep halving the interval $[a,b]$ and choose the interval where the function value on the left side is $>0$, and the function value of the right side is $<0$, to find an approximation for $c$. $\endgroup$
    – Daniel P
    Commented May 24, 2023 at 9:58

1 Answer 1

4
$\begingroup$

The main way to solve non-linear equations is using the computer. There are lots of tools that can be used if you do not need a formula for the solution.

The computer will give you some numbers $x$ that you can put into the formula you want to solve, and you will see that the numbers solves the equation, such that the left hand side equals 1000.000000 within numerical precision.

If you want a formula, the situation is different. You might use a corollary of the inverse function theorem to prove that there exists a formula that solves the equation exactly. But of course, even if it exists, it might not be practical to write it down on any sized-peace of paper because it is so large or our tools for understanding the contents is too weak. Please add comments if you want to know more.

$\endgroup$

You must log in to answer this question.