Analytic Solution for $x^2 - x \ln x -k = 0$ ??

Question

Looking for the analytic solution of $x^2 - x \ln x -k = 0$ for $x$. Have tried symbolic solver in MATLAB but couldn't find a closed-form expression. Here $0 < k < 1$.

Answer 1

You cannot find an analytical solution and you will need numerical methods.

Consider that you look for the zero(s) of function $$f(x)=x^2 - x \log(x) -k $$ $$f'(x)=2 x-\log (x)-1$$ $$f''(x)=2-\frac{1}{x}$$

The first derivative cancels at a point $$x_*=-\frac{1}{2} W\left(-\frac{2}{e}\right)$$ where appears Lambert function. But this is not a real value since, in the real domain, this function is "defined" if $x \geq -\frac 1e$ and then the first derivative is always positive.

Since the function is bounded by $g(x)=x^2-k$, you could start Newton method with $x_0=\sqrt k$ (except for $k=1$ for which $x=1$ is the solution) and generate iterates according to $$x_{n+1}=x_n-\frac{x_n^2-x_n \log (x_n)-k}{2 x_n-\log (x_n)-1}$$ and this would converge quite fast as shown below for $k=\frac12$. $$\left( \begin{array}{cc} n & x_n \\ 0 & 0.7071067812 \\ 1 & 0.3849870727 \\ 2 & 0.3633137719 \\ 3 & 0.3635195741 \\ 4 & 0.3635195956 \end{array} \right)$$ You could generate a better estimate assuming that $x^2-x\log(x)\approx x^a$ and get $a$ minimizing $$\Phi(a)=\int_0^1 \left(x^2-x\log(x)- x^a \right)^2\,dx=\frac{1}{2 a+1}-\frac{2}{(a+2)^2}-\frac{2}{a+3}+\frac{431}{1080}$$ $$\frac{d\Phi(a)}{da}=-\frac{2}{(2 a+1)^2}+\frac{4}{(a+2)^3}+\frac{2}{(a+3)^2}=0 \implies a\approx 0.7\implies x_0=k^{10/7}$$ Surprisingly, the optimal value of $a=0.692477$ is very close to $\log(2)$.

For the worked example, this would give $x_0=0.3715$ and Newton method will converge quite fast.