The muchange command receives a change of variables, an integrating factor of an ODE, and the dependent variable y(x), and it returns the integrating factor of the problem in the new variables. The change of variables is performed by making calls to dchange, and hence the same extra arguments accepted by dchange are accepted by muchange as well.

The change of variables performed by muchange is complementary to the (same) change of variables performed in the ODE in that, if $\mathrm{\mu }\left(x\,y\left(x\right)\right)$ is an integrating factor of an ODE in y(x), then their product is a total derivative,

mu*ODE = Diff(R1(x,y(x)),x);

$\mathrm{\mu }\mathrm{ODE}=\frac{\ⅆ}{\ⅆx}\mathrm{R1}\left(x\,y\left(x\right)\right)$

and then, under a change of variables

tr := {x=X(t,u(t)), y(x)=Y(t,u(t))};

$\mathrm{tr}≔\left\{x=X\left(t\,u\left(t\right)\right)\,y\left(x\right)=Y\left(t\,u\left(t\right)\right)\right\}$

where the new variables are {t, u(t)}, the following ODE is also exact:

'muchange(tr, mu, y(x)) * dchange(tr, ODE)' = Diff(R2(t,u(t)),t);

$\mathrm{muchange}\left(\mathrm{tr}\,\mathrm{\mu }\,y\left(x\right)\right)\mathrm{dchange}\left(\mathrm{tr}\,\mathrm{ODE}\right)=\frac{\ⅆ}{\ⅆt}\mathrm{R2}\left(t\,u\left(t\right)\right)$

(note however that $\mathrm{R1}\ne \mathrm{dchange}\left(\mathrm{tr}\,\mathrm{R2}\right)$).

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

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

$\mathrm{with}\left(\mathrm{PDEtools}\,\mathrm{dchange}\right)\:$

$\mathrm{ODE1}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\frac{x{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2+y\left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+1+\mathrm{diff}\left(y\left(x\right)\,x\right)}{y\left(x\right)x}=0$

$\mathrm{ODE1}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\frac{x{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2+y\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+1+\frac{\ⅆ}{\ⅆx}y\left(x\right)}{y\left(x\right)x}=0$

$\mathrm{\Μ1}≔\mathrm{intfactor}\left(\mathrm{ODE1}\right)$

$\mathrm{\Μ1}≔xy\left(x\right)$

$\mathrm{mutest}\left(\mathrm{\Μ1}\,\mathrm{ODE1}\right)$

$0$

$\mathrm{tr}≔\left\{x=\frac{1}{u\left(t\right)}\,y\left(x\right)=\frac{1}{t}\right\}$

$\mathrm{tr}≔\left\{x=\frac{1}{u\left(t\right)}\,y\left(x\right)=\frac{1}{t}\right\}$

$\mathrm{ODE2}≔\mathrm{dchange}\left(\mathrm{tr}\,\mathrm{ODE1}\,\mathrm{normal}\right)$

$\mathrm{ODE2}≔\frac{u\left(t\right)\left(t^4{\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)}^3+{u\left(t\right)}^2{\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)}^2{t}^2+\left(\frac{{\ⅆ}^2}{\ⅆt^2}u\left(t\right)\right){u\left(t\right)}^{3}t-{u\left(t\right)}^2{\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)}^{2}t+3{u\left(t\right)}^3\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)\right)}{t^3{\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)}^3}=0$

$\mathrm{\Μ2}≔\mathrm{muchange}\left(\mathrm{tr}\,\mathrm{\Μ1}\,y\left(x\right)\right)$

$\mathrm{\Μ2}≔-\frac{\frac{\ⅆ}{\ⅆt}u\left(t\right)}{{u\left(t\right)}^{3}t}$

$\mathrm{mutest}\left(\mathrm{\Μ2}\,\mathrm{ODE2}\right)$

$0$

$\mathrm{exact\_ODE}≔\mathrm{\mu }\left(x\,y\left(x\right)\right)\mathrm{ODE}\left(x\,y\left(x\right)\,\mathrm{diff}\left(y\left(x\right)\,x\right)\right)=\mathrm{diff}\left(R\left(x\right)\,x\right)$

$\mathrm{exact\_ODE}≔\mathrm{\mu }\left(x\,y\left(x\right)\right)\mathrm{ODE}\left(x\,y\left(x\right)\,\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=\frac{\ⅆ}{\ⅆx}R\left(x\right)$

$\mathrm{tr}≔\left\{x=X\left(t\,u\left(t\right)\right)\,y\left(x\right)=Y\left(t\,u\left(t\right)\right)\right\}$

$\mathrm{tr}≔\left\{x=X\left(t\,u\left(t\right)\right)\,y\left(x\right)=Y\left(t\,u\left(t\right)\right)\right\}$

$\mathrm{dchange}\left(\mathrm{tr}\,\mathrm{exact\_ODE}\,\left[t\,u\left(t\right)\right]\,\mathrm{known}=\left\{\mathrm{ODE}\,\mathrm{\mu }\right\}\right)$

$\mathrm{\mu }\left(X\left(t\,u\left(t\right)\right)\,Y\left(t\,u\left(t\right)\right)\right)\mathrm{ODE}\left(X\left(t\,u\left(t\right)\right)\,Y\left(t\,u\left(t\right)\right)\,\frac{{\mathrm{D}}_1\left(Y\right)\left(t\,u\left(t\right)\right)+{\mathrm{D}}_2\left(Y\right)\left(t\,u\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)}{{\mathrm{D}}_1\left(X\right)\left(t\,u\left(t\right)\right)+{\mathrm{D}}_2\left(X\right)\left(t\,u\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)}\right)=\frac{{\mathrm{D}}_1\left(R\right)\left(t\,u\left(t\right)\right)+{\mathrm{D}}_2\left(R\right)\left(t\,u\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)}{{\mathrm{D}}_1\left(X\right)\left(t\,u\left(t\right)\right)+{\mathrm{D}}_2\left(X\right)\left(t\,u\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)}$

$\mathrm{Diff}\left(X\left(t\,u\left(t\right)\right)\,t\right)$

$\frac{\ⅆ}{\ⅆt}X\left(t\,u\left(t\right)\right)$

$\mathrm{\μ2}\left(t\,u\left(t\right)\right)=\mathrm{Diff}\left(X\left(t\,u\left(t\right)\right)\,t\right)'\mathrm{dchange}'\left('\mathrm{tr}'\,\mathrm{\mu }\left(x\,y\right)\right)$

$\mathrm{\μ2}\left(t\,u\left(t\right)\right)=\left(\frac{\ⅆ}{\ⅆt}X\left(t\,u\left(t\right)\right)\right)\mathrm{dchange}\left(\mathrm{tr}\,\mathrm{\mu }\left(x\,y\right)\right)$

$\mathrm{\Μ1}≔\mathrm{\mu }\left(x\,\mathrm{diff}\left(y\left(x\right)\,x\right)\right)$

$\mathrm{\Μ1}≔\mathrm{\mu }\left(x\,\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

$\mathrm{ODE1}≔\mathrm{redode}\left(\mathrm{\Μ1}\,y\left(x\right)\,2\right)$

$\mathrm{ODE1}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=-\frac{{\int }_{\phantom{\mathrm{`\; `}}}^{\frac{\ⅆ}{\ⅆx}y\left(x\right)}\left(\frac{\partial }{\partial x}\mathrm{\mu }\left(x\,\mathrm{\_a}\right)\right)\ⅆ\mathrm{\_a}+{\mathrm{D}}_1\left(\mathrm{\_F1}\right)\left(x\,y\left(x\right)\right)+{\mathrm{D}}_2\left(\mathrm{\_F1}\right)\left(x\,y\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{\mathrm{\mu }\left(x\,\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}$

$\mathrm{tr}≔\left\{x=u\left(t\right)\,y\left(x\right)=t\right\}$

$\mathrm{tr}≔\left\{x=u\left(t\right)\,y\left(x\right)=t\right\}$

$\mathrm{\Μ2}≔\mathrm{muchange}\left(\mathrm{tr}\,\mathrm{\Μ1}\,y\left(x\right)\,\left[t\,u\left(t\right)\right]\right)$

$\mathrm{\Μ2}≔\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)\mathrm{\mu }\left(u\left(t\right)\,\frac{1}{\frac{\ⅆ}{\ⅆt}u\left(t\right)}\right)$

$\mathrm{ODE2}≔\mathrm{dchange}\left(\mathrm{tr}\,\mathrm{ODE1}\,\left[t\,u\left(t\right)\right]\,\mathrm{known}=\mathrm{all}\,\mathrm{normal}\right)$

$\mathrm{ODE2}≔-\frac{\frac{{\ⅆ}^2}{\ⅆt^2}u\left(t\right)}{{\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)}^3}=-\frac{\left({\int }_{\phantom{\mathrm{`\; `}}}^{\frac{1}{\frac{\ⅆ}{\ⅆt}u\left(t\right)}}{\mathrm{D}}_1\left(\mathrm{\mu }\right)\left(u\left(t\right)\,\mathrm{\_a}\right)\ⅆ\mathrm{\_a}\right)\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)+{\mathrm{D}}_1\left(\mathrm{\_F1}\right)\left(u\left(t\right)\,t\right)\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)+{\mathrm{D}}_2\left(\mathrm{\_F1}\right)\left(u\left(t\right)\,t\right)}{\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)\mathrm{\mu }\left(u\left(t\right)\,\frac{1}{\frac{\ⅆ}{\ⅆt}u\left(t\right)}\right)}$

$\mathrm{mutest}\left(\mathrm{\Μ2}\,\mathrm{ODE2}\right)$

$0$