Integrating factors for second and higher order linear ODEs

For linear ODEs (LODEs) of order 2 or greater, it is possible to calculate integrating factors by solving the adjoint of the LODE. This could be as difficult as the original problem, or much easier, depending on the example. This method is implemented in dsolve.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{adjoint}\,\mathrm{firint}\,\mathrm{intfactor}\right)\:$

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

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

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

$-\frac{\left({\ⅇ}^{x}x-1\right)y\left(x\right)}{\ln \left(x\right)x^2}+{\ⅇ}^x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)$

$\mathrm{dsolve}\left(\right)$

$y\left(x\right)=\left(\left(\int \frac{{\ⅇ}^{-{\ⅇ}^x}}{{\ln \left(x\right)}^2}\ⅆx\right)\mathrm{c\_\_1}+\mathrm{c\_\_2}\right)\ln \left(x\right)$

$\mathrm{eval}\left(\mathrm{rhs}\left(\right)\,\left\{\mathrm{\_C1}=0\,\mathrm{\_C2}=1\right\}\right)$

$\ln \left(x\right)$

$\mathrm{eval}\left(\mathrm{rhs}\left(\right)\,\left\{\mathrm{\_C1}=1\,\mathrm{\_C2}=2\right\}\right)$

$\left(\int \frac{{\ⅇ}^{-{\ⅇ}^x}}{{\ln \left(x\right)}^2}\ⅆx+2\right)\ln \left(x\right)$

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

$\mathrm{{\rm M}}≔\ln \left(x\right),\left(\int \frac{{\ⅇ}^{-{\ⅇ}^x}}{{\ln \left(x\right)}^2}\ⅆx\right)\ln \left(x\right)$

$\mathrm{firint}\left(\mathrm{{\rm M}}\left[1\right]\mathrm{ode}\right)$

$-\frac{\left(\ln \left(x\right)x^2{\ⅇ}^x+x\right)y\left(x\right)}{x^2}+\ln \left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\mathrm{c\_\_1}=0$

$\mathrm{firint}\left(\mathrm{{\rm M}}\left[2\right]\mathrm{ode}\right)$

$-\frac{\left(\left(\int \frac{{\ⅇ}^{-{\ⅇ}^x}}{{\ln \left(x\right)}^2}\ⅆx\right){\ln \left(x\right)}^2{\ⅇ}^x{x}^2+\left(\int \frac{{\ⅇ}^{-{\ⅇ}^x}}{{\ln \left(x\right)}^2}\ⅆx\right)\ln \left(x\right)x+{\ⅇ}^{-{\ⅇ}^x}{x}^2\right)y\left(x\right)}{\ln \left(x\right)x^2}+\left(\int \frac{{\ⅇ}^{-{\ⅇ}^x}}{{\ln \left(x\right)}^2}\ⅆx\right)\ln \left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\mathrm{c\_\_1}=0$

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

$y\left(x\right)=\mathrm{c\_\_1}{\ⅇ}^{{\ⅇ}^x}\ln \left(x\right)+\mathrm{c\_\_2}{\ⅇ}^{{\ⅇ}^x}\left(\int \frac{{\ⅇ}^{-{\ⅇ}^x}}{{\ln \left(x\right)}^2}\ⅆx\right)\ln \left(x\right)$