IntegratingFactor(ODE, y(x)) attempts to find an integrating factor for an ODE.

Multiplying by the integrating factor makes the ODE exact, that is, a total derivative with respect to x. The new equation can then be solved by simply integrating with respect to x.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\right)\:$

$\mathrm{ode1}≔\mathrm{diff}\left(z\left(t\right)\,t\right)\left(t-1\right){z\left(t\right)}^2=-t^2\left(z\left(t\right)+1\right)$

$\mathrm{ode1}≔\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)\left(t-1\right){z\left(t\right)}^2=-t^2\left(z\left(t\right)+1\right)$

$\mathrm{\μ1}≔\mathrm{IntegratingFactor}\left(\mathrm{ode1}\,z\left(t\right)\right)$

$\mathrm{\μ1}≔\frac{1}{\left(z\left(t\right)+1\right)\left(t-1\right)}$

$\mathrm{exact\_ode1}≔\mathrm{\μ1}\mathrm{ode1}$

$\mathrm{exact\_ode1}≔\frac{\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right){z\left(t\right)}^2}{z\left(t\right)+1}=-\frac{t^2}{t-1}$

$\mathrm{Integrate}\left(\mathrm{exact\_ode1}\,z\left(t\right)\right)$

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

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)=\frac{\left(x\mathrm{diff}\left(y\left(x\right)\,x\right)+y\left(x\right)\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)}{y\left(x\right)x}$

$\mathrm{ode2}≔\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)=\frac{\left(x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+y\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{y\left(x\right)x}$

$\mathrm{\μ2}≔\mathrm{IntegratingFactor}\left(\mathrm{ode2}\,y\left(x\right)\right)$

$\mathrm{\μ2}≔\frac{1}{y\left(x\right)x}$

$\mathrm{exact\_ode2}≔\mathrm{\μ2}\mathrm{ode2}$

$\mathrm{exact\_ode2}≔\frac{\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)}{y\left(x\right)x}=\frac{\left(x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+y\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{{y\left(x\right)}^2{x}^2}$

$\mathrm{sol}≔\mathrm{Integrate}:-\mathrm{Apply}\left(\mathrm{exact\_ode2}\,y\left(x\right)\right)$

$\mathrm{sol}≔\int \left(\frac{\left(x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+y\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{{y\left(x\right)}^2{x}^2}-\frac{\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)}{y\left(x\right)x}\right)\ⅆx=\int 0\ⅆx+\mathrm{c\_\_1}$

$\mathrm{Integrate}:-\mathrm{Evaluate}\left(\mathrm{sol}\right)$

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

The Student[ODEs][IntegratingFactor] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.