Issue solving an equation

Question

I'm at lost trying to solve the following equation : $$B\cdot x^{\frac{2}{3}}+C\cdot x^{\frac{1}{2}}=D$$

My research lead me to think that it's a transcendental equation but I don't know how to solve it...

Thanks for reading,

Regards,

76MPaul

Answer 1

It can help to reduce the number of parameters. Dividing by $D$, you get

$$bx^{2/3}+cx^{1/2}=1.$$

Then, with $x:=a^6/z^6$,

$$b\frac{a^4}{z^4}+c\frac{a^3}{z^3}=1$$

becomes

$$z^4-z=s$$ by setting $ca^3=1$.

This gives a little more insight in the function (the inverse of $f(z)=z^4-z$) and makes the analytical expression a little more "human friendly".

http://www.wolframalpha.com/input/?i=solve+t%5E4-t%3Ds+for+t

In fact, by setting $g(s):=\sqrt[3]{\sqrt 3\sqrt{256s^3+27}+9}$, you really get something "manageable", like

$$2z=-\sqrt{\frac{g(s)}{\sqrt[3]2\sqrt{\frac23}}-\frac{4\sqrt[3]\frac23s}{g(s)}}-\sqrt{\frac{4\sqrt[3]\frac23s}{g(s)}-\frac{g(s)}{\sqrt[3]2\sqrt{\frac23}}-\dfrac2{\sqrt{\frac{g(s)}{\sqrt[3]2\sqrt{\frac23}}-\frac{4\sqrt[3]\frac23s}{g(s)}}}}$$

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Note that I chose the above form in case you would try to solve the quartic by hand: it is already depressed.

Answer 2

There is a multivalued function called Lambert-Tsallis https://doi.org/10.1016/j.physa.2019.03.046, $W_q(z)$. It is the solution of $ W_q(z) \cdot \bigg( 1 + (1-q)\cdot W_q(z) \bigg)^\frac{1}{1-q} = z$

Your problem concerns noninteger polynomys and even with some manipulations proposed by anyone, it is almost sure you will have to test or neglect some solution in the moment you change variables. Using the formula below with $W_q(z)$ function is not different. However, following you have the numbers that you can test yourself.

Your solution can be written as $x=\bigg [-\frac{4B}{C} \cdot W_{5}\bigg(-{\left (\frac{D}{B}\right )}^{-0.25}\cdot \frac{C}{4B} \bigg) \bigg]^{-6}$. A few values are provided below

\begin{matrix} B & C & D & z & W_5(z) & x\\ +1.0 & +2.0 & +4.0 & -0.3536 & (-0.45877 , -0.00000i) & (+1.67587 , -0.00000i) \\ +1.0 & -2.0 & +4.0 & +0.3536 & (+0.21550 , -0.00000i) & (+156.02316 , +0.00000i) \\ -5.0 & -2.0 & -4.0 & -0.1057 & (-0.11634 , -0.00000i) & (+0.40339 , -0.00000i) \\ \pi & \sqrt{2} & 10 & -0.0843 & (-0.09106 , -0.00000i) & (+3.56405 , -0.00000i) \\ \end{matrix}

Based on my tests, I detected that are somes conditions among B, C and D. Here I have considered consistent data, for sake of simplicity.