The mutest command checks whether a given expression is an integrating factor of a given ODE. Similar to odetest, mutest returns $0$ when the first argument - say mu - is indeed an integrating factor.  Otherwise it returns an algebraic expression obtained after simplifying the Exactness condition. This Exactness condition is obtained by applying Euler's operator (satisfied by total derivatives) to mu times the ODE.

Note however that when the result returned by mutest is not zero, the expression might nevertheless be an integrating factor. In some cases you can obtain the desired $0$ with further simplifications by using commands such as expand and combine.

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

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

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=y\left(x\right)a\left(x\right)+b\left(x\right)$

$\mathrm{ODE}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=y\left(x\right)a\left(x\right)+b\left(x\right)$

$\mathrm{{\rm M}}≔\mathrm{intfactor}\left(\mathrm{ODE}\right)$

$\mathrm{{\rm M}}≔{\ⅇ}^{\int -a\left(x\right)\ⅆx}$

$\mathrm{mutest}\left(\mathrm{{\rm M}}\,\mathrm{ODE}\right)$

$0$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=-\frac{{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2+2x\mathrm{diff}\left(y\left(x\right)\,x\right)+2y\left(x\right)}{y\left(x\right)}$

$\mathrm{ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=-\frac{{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2+2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+2y\left(x\right)}{y\left(x\right)}$

$\mathrm{{\rm M}}≔\frac{1}{\mathrm{diff}\left(y\left(x\right)\,x\right)+2x}$

$\mathrm{{\rm M}}≔\frac{1}{\frac{\ⅆ}{\ⅆx}y\left(x\right)+2x}$

$\mathrm{mutest}\left(\mathrm{{\rm M}}\,\mathrm{ODE}\right)$

$0$