Can we use the Lambert W solution $y=-3W(K_2x^{-4/3})$ instead of $y =-3W(\frac 1 3\sqrt[3]{-\frac{K_1}{x^4}})$ if we choose an appropriate constant?

During the process of solving the separable differential equation $$4y - x(y-3)y' = 0$$, our solution acquires a constant when we go from $$\frac 4 x = \frac{y-3}{y}y'$$ to $$\ln x + C_1 = y - 3 \ln y$$. We then do: $$e^{-\frac{4}{3}(\ln x) -\frac{C_1}{3}} = e^{\ln y - \frac y 3}$$ $$x^{-\frac 4 3}e^{-\frac{C_1}3}= ye^{-\frac y 3}$$ $$-\frac 1 3 x^{-\frac 4 3}e^{-\frac{C_1}3}= -\frac y 3 e^{-\frac y 3}$$ I substitute $$- \frac 1 3 e^{\frac {C_1}3}$$ by the constant $$K_1$$ to get $$K_1x^{-\frac 4 3}=-\frac y 3 e^{-\frac y 3}$$ applying the Lambert W function I get: $$-3W(K_1x^{-\frac 4 3}) = y$$

Instead, wolframalpha suggests me the solution $$y = -3W\left( \frac 1 3\sqrt[3]{-\frac{K_2}{x^4}}\right)$$.

My question is: is the process through which I substitute in the constant $$K_1$$ legitimate? Since otherwise, I'm unable to determine why my answer doesn't correspond to the wolframalpha one. The reason why I thought I was allowed to substitute $$K_1$$ was because, in the step where we go from $$\frac 4 x = \frac{y-3}{y}y'$$ to $$\ln x + C_1 = y - 3 \ln y$$, we obviously see that taking the derivative of any constant makes our equation true, and therefore we would be free to choose whichever $$K_1$$ makes our equation look nice. But is this reasoning correct, given that wolframalpha provides a more complicated solution?

Your answer is correct, and comparing the two constants we have $$K_{1}=-\frac{1}{3}K_{2}^\frac{1}{3}$$.

After integration you can also define the constant of integration as follows

$$\frac{4}{x}=\frac{y-3}{y}y'$$ $$4\ln(x)-\ln(K_2)=y-3\ln(y)$$ $$\dots$$ $$-\frac{y}{3}e^{-\frac{y}{3}}=\frac{1}{3}\sqrt[3]{-\frac{K_2}{x^4}}$$

Or instead of setting $$K_{1}=-\frac{1}{3}e^{-\frac{C_1}{3}}$$, set $$K_{2}=e^{-C_{1}}$$, to obtain the solution from WA.

• Thank you for your answer! Is there no risk of odd behaviour in the complex plane which could make an attempt to equate $K_1$ to a function of $K_2$ restricted to a specific domain given that our choice of constant could make possible complex roots for the cube root? Oct 24, 2021 at 8:57
• Consider, e.g., that $\sqrt[3]{-\frac{K_2}{x^4}}$ could be ambiguous w.r.t. which of $-K_2$ or $-x^4$ is the negative term, and so potentially ambiguous w.r.t. which term determines the existence of complex roots within the cube root Oct 24, 2021 at 9:02
• Thanks for your comment, note that both $K_1$ and $K_2$ are constants, so this does not change the final answer. The minus sign can be taken outside (but still inside the brackets) since the cube root is a bijective function. What is true is that there are more solutions from WA. All we did is define the constant of integration differently. Oct 24, 2021 at 9:57
• Thank you for the help! Oct 24, 2021 at 10:27
• You're very welcome! :-) Oct 24, 2021 at 13:55