The splitsys function returns the received set of equations split, as much as possible, into subsets of equations mutually coupled, but not coupled with the equations of the other subsets. This splitting of the set may be useful for solving the system.

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

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

$\mathrm{sys}≔\left\{k\left(-2\mathrm{q2}\left(t\right)+2\mathrm{q3}\left(t\right)\right)=-\mathrm{diff}\left(\mathrm{p3}\left(t\right)\,t\right)\,k\left(2\mathrm{q2}\left(t\right)-2\mathrm{q3}\left(t\right)\right)=-\mathrm{diff}\left(\mathrm{p2}\left(t\right)\,t\right)\,2k\mathrm{q1}\left(t\right)=-\mathrm{diff}\left(\mathrm{p1}\left(t\right)\,t\right)\,\mathrm{p3}\left(t\right)\mathrm{\alpha }=\mathrm{diff}\left(\mathrm{q2}\left(t\right)\,t\right)\,\mathrm{p2}\left(t\right)\mathrm{\alpha }+\frac{\mathrm{p3}\left(t\right)}{\mathrm{m3}}=\mathrm{diff}\left(\mathrm{q3}\left(t\right)\,t\right)\,\frac{\mathrm{p1}\left(t\right)}{\mathrm{m1}}=\mathrm{diff}\left(\mathrm{q1}\left(t\right)\,t\right)\right\}$

$\mathrm{sys}≔\left\{k\left(-2\mathrm{q2}\left(t\right)+2\mathrm{q3}\left(t\right)\right)=-\frac{\ⅆ}{\ⅆt}\mathrm{p3}\left(t\right)\,k\left(2\mathrm{q2}\left(t\right)-2\mathrm{q3}\left(t\right)\right)=-\frac{\ⅆ}{\ⅆt}\mathrm{p2}\left(t\right)\,\frac{\mathrm{p1}\left(t\right)}{\mathrm{m1}}=\frac{\ⅆ}{\ⅆt}\mathrm{q1}\left(t\right)\,\mathrm{p3}\left(t\right)\mathrm{\alpha }=\frac{\ⅆ}{\ⅆt}\mathrm{q2}\left(t\right)\,2k\mathrm{q1}\left(t\right)=-\frac{\ⅆ}{\ⅆt}\mathrm{p1}\left(t\right)\,\mathrm{p2}\left(t\right)\mathrm{\alpha }+\frac{\mathrm{p3}\left(t\right)}{\mathrm{m3}}=\frac{\ⅆ}{\ⅆt}\mathrm{q3}\left(t\right)\right\}$

$\mathrm{funcs}≔\mathrm{indets}\left(\mathrm{sys}\,\mathrm{Function}\right)$

$\mathrm{funcs}≔\left\{\mathrm{p1}\left(t\right)\,\mathrm{p2}\left(t\right)\,\mathrm{p3}\left(t\right)\,\mathrm{q1}\left(t\right)\,\mathrm{q2}\left(t\right)\,\mathrm{q3}\left(t\right)\right\}$

$\mathrm{splitsys}\left(\mathrm{sys}\,\mathrm{funcs}\right)$

$\left\{\frac{\mathrm{p1}\left(t\right)}{\mathrm{m1}}=\frac{\ⅆ}{\ⅆt}\mathrm{q1}\left(t\right)\,2k\mathrm{q1}\left(t\right)=-\frac{\ⅆ}{\ⅆt}\mathrm{p1}\left(t\right)\right\},\left\{k\left(-2\mathrm{q2}\left(t\right)+2\mathrm{q3}\left(t\right)\right)=-\frac{\ⅆ}{\ⅆt}\mathrm{p3}\left(t\right)\,k\left(2\mathrm{q2}\left(t\right)-2\mathrm{q3}\left(t\right)\right)=-\frac{\ⅆ}{\ⅆt}\mathrm{p2}\left(t\right)\,\mathrm{p3}\left(t\right)\mathrm{\alpha }=\frac{\ⅆ}{\ⅆt}\mathrm{q2}\left(t\right)\,\mathrm{p2}\left(t\right)\mathrm{\alpha }+\frac{\mathrm{p3}\left(t\right)}{\mathrm{m3}}=\frac{\ⅆ}{\ⅆt}\mathrm{q3}\left(t\right)\right\}$

$\mathrm{sys}≔\left\{x=r\left(x\right)\cos \left(\mathrm{\phi }\left(x\right)\right)\,\mathrm{diff}\left(f\left(x\right)\,x\right)=\cos \left(g\left(x\right)\right)\,\mathrm{diff}\left(g\left(x\right)\,x\right)=f\left(x\right)\,y\left(x\right)=r\left(x\right)\sin \left(\mathrm{\phi }\left(x\right)\right)\right\}$

$\mathrm{sys}≔\left\{x=r\left(x\right)\cos \left(\mathrm{\phi }\left(x\right)\right)\,\frac{\ⅆ}{\ⅆx}f\left(x\right)=\cos \left(g\left(x\right)\right)\,\frac{\ⅆ}{\ⅆx}g\left(x\right)=f\left(x\right)\,y\left(x\right)=r\left(x\right)\sin \left(\mathrm{\phi }\left(x\right)\right)\right\}$

$\mathrm{splitsys}\left(\mathrm{sys}\,\left\{f\,g\,\mathrm{\phi }\,r\right\}\left(x\right)\right)$

$\left\{x=r\left(x\right)\cos \left(\mathrm{\phi }\left(x\right)\right)\,y\left(x\right)=r\left(x\right)\sin \left(\mathrm{\phi }\left(x\right)\right)\right\},\left\{\frac{\ⅆ}{\ⅆx}f\left(x\right)=\cos \left(g\left(x\right)\right)\,\frac{\ⅆ}{\ⅆx}g\left(x\right)=f\left(x\right)\right\}$

$\mathrm{PDE\_sys}≔\left\{{\mathrm{diff}\left(g\left(x\,y\right)\,x\right)}^2=g\left(x\,y\right)-f\left(x\,y\right)\,\mathrm{diff}\left(f\left(x\,y\right)\,x\right)=g\left(x\,y\right)+f\left(x\,y\right)\,\mathrm{diff}\left(h\left(x\,y\,z\right)\,x\right)=zk\left(x\,y\right)\right\}$

$\mathrm{PDE\_sys}≔\left\{{\left(\frac{\partial }{\partial x}g\left(x\,y\right)\right)}^2=g\left(x\,y\right)-f\left(x\,y\right)\,\frac{\partial }{\partial x}f\left(x\,y\right)=g\left(x\,y\right)+f\left(x\,y\right)\,\frac{\partial }{\partial x}h\left(x\,y\,z\right)=zk\left(x\,y\right)\right\}$

$\mathrm{splitsys}\left(\mathrm{PDE\_sys}\,\left\{f\left(x\,y\right)\,g\left(x\,y\right)\,h\left(x\,y\,z\right)\,k\left(x\,y\right)\right\}\right)$

$\left\{\frac{\partial }{\partial x}h\left(x\,y\,z\right)=zk\left(x\,y\right)\right\},\left\{{\left(\frac{\partial }{\partial x}g\left(x\,y\right)\right)}^2=g\left(x\,y\right)-f\left(x\,y\right)\,\frac{\partial }{\partial x}f\left(x\,y\right)=g\left(x\,y\right)+f\left(x\,y\right)\right\}$