This is a first order nonlinear differential equation. Can this ODE be solved for $x(z)$? $$ \frac{\mathrm{d}x}{\mathrm{d}z}= ax^5+bx+c $$
I have tried using a variable substitution but it did not work: any ideas?
This DE is found after solving a more complicated system of DEs; it was part of my physics research to find soliton solutions of a system of DEs. If a,b,c were known then I could use the partial fraction decomposition, but this is not the case. I was wondering if there is perhaps a physicist who has seen before such DEs.
I found out that it is Chini DE : $dy/dx=f(x)y^n-g(x)y+h(x)$. For n=2 it is Ricatti and for n=3 Abel DE. If Chini Invariant does not depend on x then it can be solved since my f,g and h functions are constant. I need to find out how? There is a book of Kamke which will be helpful. Anyone knows the name of this textbook ? Thanks guys.
The only way I see would be to rewrite $$\frac{dz}{dx}=\frac 1{ ax^5+bx+c}=\frac 1a\frac 1{(x-r_1)(x-r_2)(x-r_3)(x-r_4)(x-r_5)}$$ where the $r_i$ are the roots of $ax^5+bx+c=0$. Then partial fraction decomposition to get $$\frac{dz}{dx}=\frac 1 a \sum_{i=1}^5 \frac {A_i}{x-r_i}$$ and integrating $$z+C =\frac 1 a \sum_{i=1}^5 A_i\,\log(x-r_i)$$
Not very funny !