The bernoullisol routine determines if the first argument is a first order ODE of Bernoulli type and, if so, returns a solution to the Bernoulli equation.

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 bernoullisol(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[bernoullisol](..).

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

$\mathrm{ode}≔\mathrm{diff}\left(z\left(t\right)\,t\right)+p\left(t\right)z\left(t\right)=q\left(t\right){z\left(t\right)}^2$

$\mathrm{ode}≔\frac{\ⅆ}{\ⅆt}z\left(t\right)+p\left(t\right)z\left(t\right)=q\left(t\right){z\left(t\right)}^2$

$\mathrm{bernoullisol}\left(\mathrm{ode}\,z\left(t\right)\right)$

$\left\{z\left(t\right)=\frac{{\ⅇ}^{\int -p\left(t\right)\ⅆt}}{\int -{\ⅇ}^{\int -p\left(t\right)\ⅆt}q\left(t\right)\ⅆt+\mathrm{c\_\_1}}\right\}$

$\mathrm{ode}≔\mathrm{diff}\left(z\left(t\right)\,t\right)+tz\left(t\right)=q\left(t\right){z\left(t\right)}^n\:$

$\mathrm{bernoullisol}\left(\mathrm{ode}\,z\left(t\right)\right)$

$\left\{z\left(t\right)=\frac{{\ⅇ}^{-\frac{t^{2}n}{2\left(n-1\right)}}{\ⅇ}^{\frac{t^2}{2\left(n-1\right)}}}{{\left(-n\left(\int {\ⅇ}^{-\frac{t^{2}n}{2}}{\ⅇ}^{\frac{t^2}{2}}q\left(t\right)\ⅆt\right)+\mathrm{c\_\_1}+\int {\ⅇ}^{-\frac{t^{2}n}{2}}{\ⅇ}^{\frac{t^2}{2}}q\left(t\right)\ⅆt\right)}^{\frac{1}{n-1}}}\right\}$

$\mathrm{ode}≔\mathrm{D}\left(z\right)\left(t\right)+tz\left(t\right)=t{z\left(t\right)}^n\:$

$\mathrm{bernoullisol}\left(\mathrm{ode}\,z\left(t\right)\right)$

$\left\{z\left(t\right)=\frac{1}{{\left(1+{\ⅇ}^{\frac{t^2\left(n-1\right)}{2}}\mathrm{c\_\_1}\right)}^{\frac{1}{n-1}}}\right\}$