The varparam routine is used to find a general solution of an ordinary differential equation (ODE) by the method of variation of parameters.  In the event that no solution form can be generated (that is, if the Wronskian of the solution list sols is zero), then FAIL is returned. Otherwise, an general solution is generated, with indexed global constants _C.

The list sols must be a list of linearly independent solutions to a corresponding homogeneous ODE. The second argument, v, is the right-hand-side of a linear ODE where it is assumed that the coefficient of the highest derivative is equal to one. If this right-hand-side is zero, then 0 is the only particular solution.

If a fourth argument true is added, the routine returns only the particular solution.

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

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

$\mathrm{varparam}\left(\left[\mathrm{u1}\left(x\right)\,\mathrm{u2}\left(x\right)\right]\,g\left(x\right)\,x\right)$

${\mathrm{\_C}}_1\mathrm{u1}\left(x\right)+{\mathrm{\_C}}_2\mathrm{u2}\left(x\right)-\left(\int \frac{\mathrm{u2}\left(x\right)g\left(x\right)}{\mathrm{u1}\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{u2}\left(x\right)\right)-\mathrm{u2}\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{u1}\left(x\right)\right)}\ⅆx\right)\mathrm{u1}\left(x\right)+\left(\int \frac{\mathrm{u1}\left(x\right)g\left(x\right)}{\mathrm{u1}\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{u2}\left(x\right)\right)-\mathrm{u2}\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{u1}\left(x\right)\right)}\ⅆx\right)\mathrm{u2}\left(x\right)$

$\mathrm{varparam}\left(\left[\mathrm{u1}\left(x\right)\,\mathrm{u2}\left(x\right)\right]\,g\left(x\right)\,x\,\mathrm{true}\right)$

$-\left(\int \frac{\mathrm{u2}\left(x\right)g\left(x\right)}{\mathrm{u1}\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{u2}\left(x\right)\right)-\mathrm{u2}\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{u1}\left(x\right)\right)}\ⅆx\right)\mathrm{u1}\left(x\right)+\left(\int \frac{\mathrm{u1}\left(x\right)g\left(x\right)}{\mathrm{u1}\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{u2}\left(x\right)\right)-\mathrm{u2}\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{u1}\left(x\right)\right)}\ⅆx\right)\mathrm{u2}\left(x\right)$

$\mathrm{de}≔\mathrm{diff}\left(Y\left(x\right)\,x\,x\right)+Y\left(x\right)\:$

$\mathrm{sols}≔\mathrm{dsolve}\left(\mathrm{de}\,Y\left(x\right)\,\mathrm{output}=\mathrm{basis}\right)$

$\mathrm{sols}≔\left[\sin \left(x\right)\,\cos \left(x\right)\right]$

$\mathrm{varparam}\left(\mathrm{sols}\,\csc \left(x\right)\,x\right)$

${\mathrm{\_C}}_1\sin \left(x\right)+{\mathrm{\_C}}_2\cos \left(x\right)-\ln \left(\csc \left(x\right)\right)\sin \left(x\right)-x\cos \left(x\right)$

$\mathrm{dsolve}\left(\mathrm{de}=\csc \left(x\right)\,Y\left(x\right)\right)$

$Y\left(x\right)=\sin \left(x\right)\mathrm{c\_\_2}+\cos \left(x\right)\mathrm{c\_\_1}-\ln \left(\csc \left(x\right)\right)\sin \left(x\right)-x\cos \left(x\right)$

$\mathrm{Rhs}≔x\exp \left(x\right)+2x\:$

$\mathrm{sols}≔\left[\exp \left(x\right)\,\exp \left(2x\right)\right]\:$

$\mathrm{varparam}\left(\mathrm{sols}\,\mathrm{Rhs}\,x\,\mathrm{true}\right)$

$-\frac{\left({\ⅇ}^{3x}{x}^2+2x{\ⅇ}^{3x}-2{\ⅇ}^{2x}x+2{\ⅇ}^{3x}-3{\ⅇ}^{2x}\right){\ⅇ}^{-2x}}{2}$

$\mathrm{varparam}\left(\left[\exp \left(x\right)\,\exp \left(2x\right)\,\exp \left(-x\right)\right]\,\exp \left(x\right)\,x\right)$

${\mathrm{\_C}}_1{\ⅇ}^x+{\mathrm{\_C}}_2{\ⅇ}^{2x}+{\mathrm{\_C}}_3{\ⅇ}^{-x}-\frac{x{\ⅇ}^x}{2}-\frac{{\ⅇ}^x}{4}$