See the Chini command for the format of the Chini ODE.

The chinisol routine determines if the first argument is a first order Chini ODE and, if so, attempts to find a solution to the equation by using the associated methods.

The first argument is a differential equation in diff or D form and the second argument is the function in the differential equation.

This function is part of the DEtools package, and so it can be used in the form chinisol(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[chinisol](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=a{y\left(x\right)}^5+\frac{b}{x^{\frac{5}{4}}}$

$\mathrm{ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=a{y\left(x\right)}^5+\frac{b}{x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$

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

$\left\{y\left(x\right)=-\frac{4\mathrm{RootOf}\left(-4\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{1024{\mathrm{\_a}}^{5}a{b}^4+\mathrm{\_a}-1}\ⅆ\mathrm{\_a}\right)+\ln \left(x\right)+4\mathrm{c\_\_1}\right)b}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}\right\}$

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

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

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

$\left\{y\left(x\right)=\frac{\mathrm{RootOf}\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{{\left({f\left(x\right)}^{1-n}\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right){\left(ag\left(x\right)+b\right)}^{-n}\right)}^{-n-1}{\left(f\left(x\right)\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right)\right)}^{-2n+1}{\left({\left(ag\left(x\right)+b\right)}^{-n-1}{f\left(x\right)}^{2-n}an{\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right)}^3\right)}^n{n}^{-n}}{-\mathrm{\_a}{\left({f\left(x\right)}^{1-n}\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right){\left(ag\left(x\right)+b\right)}^{-n}\right)}^{-n-1}{\left(f\left(x\right)\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right)\right)}^{-2n+1}{\left({\left(ag\left(x\right)+b\right)}^{-n-1}{f\left(x\right)}^{2-n}an{\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right)}^3\right)}^n{n}^{-n}+{\left({f\left(x\right)}^{1-n}\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right){\left(ag\left(x\right)+b\right)}^{-n}\right)}^{-n-1}{\left(f\left(x\right)\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right)\right)}^{-2n+1}{\left({\left(ag\left(x\right)+b\right)}^{-n-1}{f\left(x\right)}^{2-n}an{\left(\frac{\ⅆ}{\ⅆx}g\left(x\right)\right)}^3\right)}^n{n}^{-n}+{\mathrm{\_a}}^n}\ⅆ\mathrm{\_a}-\ln \left(ag\left(x\right)+b\right)+\mathrm{c\_\_1}\right)f\left(x\right)\left(ag\left(x\right)+b\right)}{a}\right\}$