Integration and elliptic functions

Question

I am trying to solve the following differential equation

$$ \left( \frac{d u}{d \phi} \right)^{2} = Q u^{3} + P u^{2} + c $$

For this, I wrote it as

$$ \left( \frac{d u}{d \phi} \right)^{2} = Q u^{3} + P u^{2} + c = Q (u - \alpha) (u - \beta) (u - \gamma)$$

where

$$ \alpha + \beta + \gamma = - \frac{P}{Q} $$

$$ \alpha \gamma + \beta \gamma + \alpha \beta = 0 $$

$$ \alpha \beta \gamma = - \frac{c}{Q} $$

From now on, I took $\alpha > \beta \geq u \geq \gamma > 0 $. Therefore, the differential equation reduces to

$$ \int \frac{du}{\sqrt{ Q (u - \alpha) (u - \beta) (u - \gamma) }} = \int \phi $$

Which is equivalent to writing

$$ 1 = \int \frac{du}{\sqrt{ Q \phi (u - \alpha) (u - \beta) (u - \gamma) }} $$

For convenience I put the $\phi$ inside the integral since it is a constant. I hope to obtain a solution expressed in elliptic functions. However, I must first know $\alpha$, $\beta$ and $\gamma$. I solved the system of equations using:

Solve[{\[Alpha] + \[Beta] + \[Gamma] == -(P/
     Q), \[Alpha]*\[Gamma] + \[Beta]*\[Gamma] + \[Alpha]*\[Beta] == 
    0, \[Alpha]*\[Beta]*\[Gamma] == -(c/
     Q)}, {\[Alpha], \[Beta], \[Gamma]}];

where from:

However, when I try to integrate with the following code, I cannot understand the solution obtained. Is there an error? Or how should I proceed?

Assuming[\[Alpha] > \[Beta] > \[Gamma], 
 Integrate[1/Sqrt[
   Q^2*\[Phi]*(u - \[Alpha])*(u - \[Beta])*(u - \[Gamma])], u] // 
  Simplify]

Could someone guide me or correct any errors? Thanks in advance :)

Answer 1

Id you want a 1 on the right side you have

$$\int \frac{du}{\sqrt{Q\ (u-\alpha) \ (u-\beta) \ (u-\gamma)}} =\int d\phi = \phi(u)$$

equivalent to

$$\frac{1}{\phi(u)} \ \int \frac{du}{\sqrt{Q\ (u-\alpha) \ (u-\beta) \ (u-\gamma)}} = 1$$

If somebody puts the factor $\frac{1}{\phi(u)}$ not only inside integral as a constant independent of $u$ but also under the square root he is seeking a solution with $\phi=1$.

Most probably there is a misunderstanding of the inverse function notation

$$\phi = \Phi(u), u= \Phi^{-1}(\phi)$$

with $\Phi$ representing the elliptic integral

$$\Phi^2 \ = \ \left(\int \frac{1}{\sqrt{Q (u-\alpha ) (u-\beta ) (u-\gamma )}} \, du\right)^2 \ = \ \frac{4}{Q (\beta -\alpha )} \quad F\left(\sin ^{-1}\left(\frac{\sqrt{\beta -\alpha }}{\sqrt{u-\alpha }}\right)|\frac{\alpha -\gamma }{\alpha -\beta }\right)^2 $$

which has many different algebraic forms, that depend on the position of the three roots and the range of u anywhere between the four complex branch points including $\infty$, eg

$$-\infty \lt \alpha \lt \beta \lt \gamma \lt u \lt \infty$$ for the case that u is growing montonously for a rotating system.