The general form of Chini's equation is given by:

Chini_ode := diff(y(x),x)=f(x)*y(x)^n-g(x)*y(x)+h(x);

$\mathrm{Chini\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=f\left(x\right){y\left(x\right)}^n-g\left(x\right)y\left(x\right)+h\left(x\right)$

where f(x), g(x) and h(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, ODE 1.55, p. 303.

There is as yet no general solution for this ODE. For n=2, Chini's ODE is of Riccati type; for n=3 it is of Abel type. If the following combination of coefficients

Chini_invariant := f(x)^(-n-1)*h(x)^(-2*n+1)*(f(x)*
diff(h(x),x)-diff(f(x),x)*h(x)-g(x)*f(x)*n*h(x))^n*n^(-n);

$\mathrm{Chini\_invariant}≔{f\left(x\right)}^{-n-1}{h\left(x\right)}^{-2n+1}{\left(f\left(x\right)\left(\frac{\ⅆ}{\ⅆx}h\left(x\right)\right)-\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)h\left(x\right)-g\left(x\right)f\left(x\right)nh\left(x\right)\right)}^n{n}^{-n}$

is independent of 'x', then the solution to the ODE follows in a straightforward manner; see Kamke, page 303. This scheme, proposed by Chini, generalizes the method of invariants for Abel ODEs (also proposed by Chini) found in Kamke's book as sub-method (g) for Abel ODEs; see odeadvisor,Abel.

Actually, when the square term in the Abel ODE is zero, the Abel invariant described in odeadvisor,Abel is equal to the Chini invariant described below. Now all Abel ODEs can be rewritten in Chini format (that is, square term = 0; see example below), and if the Abel invariant of an Abel ODE is constant, then the Abel invariant for this ODE written in Chini format is also constant. To understand this, note that the "independence of the invariant with respect to 'x'" for Abel ODEs is preserved under transformations of the form

y -> G(t)*u(t)+H(t);

$y\mapsto G\left(t\right)\cdot u\left(t\right)+H\left(t\right)$

x -> F(t);

$x\mapsto F\left(t\right)$

for any G, H, and F. The transformation that removes the square term in Abel ODEs (that is, rewrites it in Chini format) is given by

y -> u(x)-f2/(f3*3);

$y\mapsto u\left(x\right)-\frac{\mathrm{f2}}{3\cdot \mathrm{f3}}$

and is just a particular case of the general transformation displayed above (see also odeadvisor,Abel.

$\mathrm{with}\left(\mathrm{PDEtools}\,\mathrm{dchange}\right)$

$\left[\mathrm{dchange}\right]$

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{odeadvisor}\left(\mathrm{Chini\_ode}\right)$

$\left[\mathrm{\_Chini}\right]$

$\mathrm{Abel\_ode}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=\mathrm{f3}\left(x\right){y\left(x\right)}^3+\mathrm{f2}\left(x\right){y\left(x\right)}^2+\mathrm{f1}\left(x\right)y\left(x\right)+\mathrm{f0}\left(x\right)$

$\mathrm{Abel\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\mathrm{f3}\left(x\right){y\left(x\right)}^3+\mathrm{f2}\left(x\right){y\left(x\right)}^2+\mathrm{f1}\left(x\right)y\left(x\right)+\mathrm{f0}\left(x\right)$

$\mathrm{tr}≔\left\{y\left(x\right)=u\left(x\right)-\frac{\mathrm{f2}\left(x\right)}{3\mathrm{f3}\left(x\right)}\right\}$

$\mathrm{tr}≔\left\{y\left(x\right)=u\left(x\right)-\frac{\mathrm{f2}\left(x\right)}{3\mathrm{f3}\left(x\right)}\right\}$

$\mathrm{Abel\_ode\_Chini}≔\mathrm{collect}\left(\mathrm{diff}\left(u\left(x\right)\,x\right)=\mathrm{solve}\left(\mathrm{dchange}\left(\mathrm{tr}\,\mathrm{Abel\_ode}\,\left[u\right]\right)\,\mathrm{diff}\left(u\left(x\right)\,x\right)\right)\,u\left(x\right)\right)$

$\mathrm{Abel\_ode\_Chini}≔\frac{\ⅆ}{\ⅆx}u\left(x\right)=\mathrm{f3}\left(x\right){u\left(x\right)}^3+\frac{\left(27{\mathrm{f3}\left(x\right)}^2\mathrm{f1}\left(x\right)-9\mathrm{f3}\left(x\right){\mathrm{f2}\left(x\right)}^2\right)u\left(x\right)}{27{\mathrm{f3}\left(x\right)}^2}+\frac{27\mathrm{f0}\left(x\right){\mathrm{f3}\left(x\right)}^2-9\mathrm{f3}\left(x\right)\mathrm{f2}\left(x\right)\mathrm{f1}\left(x\right)+2{\mathrm{f2}\left(x\right)}^3+9\left(\frac{\ⅆ}{\ⅆx}\mathrm{f2}\left(x\right)\right)\mathrm{f3}\left(x\right)-9\mathrm{f2}\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{f3}\left(x\right)\right)}{27{\mathrm{f3}\left(x\right)}^2}$

$\mathrm{coeff}\left(\mathrm{rhs}\left(\mathrm{Abel\_ode\_Chini}\right)\,u\left(x\right)\,2\right)$

$0$

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=a{y\left(x\right)}^n+b{x}^{\frac{n}{1-n}}$

$\mathrm{ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=a{y\left(x\right)}^n+b{x}^{\frac{n}{-n+1}}$

$\mathrm{odeadvisor}\left(\mathrm{ode}\right)$

$\left[\left[\mathrm{\_homogeneous}\,\mathrm{class\; G}\right]\,\mathrm{\_Chini}\right]$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\mathrm{ans}≔{\int }_{\mathrm{\_b}}^{y\left(x\right)}\frac{x^{\frac{n}{n-1}}}{\left(ax\left(n-1\right){\mathrm{\_a}}^n+\mathrm{\_a}\right)x^{\frac{n}{n-1}}+bx\left(n-1\right)}\ⅆ\mathrm{\_a}-\mathrm{c\_\_1}=0$

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,x\right)-a^n{f\left(x\right)}^{1-n}\mathrm{diff}\left(g\left(x\right)\,x\right){y\left(x\right)}^n-\frac{\mathrm{diff}\left(f\left(x\right)\,x\right)}{f\left(x\right)}y\left(x\right)-f\left(x\right)\mathrm{diff}\left(g\left(x\right)\,x\right)$

$\mathrm{ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)-a^n{f\left(x\right)}^{-n+1}\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right){y\left(x\right)}^n-\frac{\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)y\left(x\right)}{f\left(x\right)}-f\left(x\right)\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right)$

$\mathrm{odeadvisor}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left[\mathrm{\_Chini}\,\left[\mathrm{\_1st\_order}\,\mathrm{\_with\_symmetry\_[F(x),G(x)*y+H(x)]}\right]\right]$

$\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\frac{ay\left(x\right)\mathrm{LerchPhi}\left(-{\left(\frac{ay\left(x\right)}{f\left(x\right)}\right)}^n\,1\,\frac{1}{n}\right)}{nf\left(x\right)}-ag\left(x\right)+\mathrm{c\_\_1}=0$