The IVPsol command specializes the integration constants of the form $\mathrm{\_Cn}$ (where $n$ is an integer) by entering the given solution sol to match the given initial or boundary conditions ics.

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

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

$\mathrm{ode}\left[1\right]≔x\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+4{y\left(x\right)}^2=0$

${\mathrm{ode}}_1≔x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+4{y\left(x\right)}^2=0$

$\mathrm{ic}\left[1\right]≔y\left(1\right)=-\frac{1}{2},\mathrm{D}\left(y\right)\left(1\right)=\frac{1}{2}$

${\mathrm{ic}}_1≔y\left(1\right)=-\frac{1}{2},\mathrm{D}\left(y\right)\left(1\right)=\frac{1}{2}$

$\mathrm{sol}\left[1\right]≔y\left(x\right)=\frac{\mathrm{\_C1}}{2x}$

${\mathrm{sol}}_1≔y\left(x\right)=\frac{\mathrm{\_C1}}{2x}$

$\mathrm{DEtools}\left[\mathrm{IVPsol}\right]\left(\left[\mathrm{ic}\left[1\right]\right]\,\mathrm{sol}\left[1\right]\right)$

$y\left(x\right)=-\frac{1}{2x}$

$\mathrm{ode}\left[2\right]≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-2{y\left(x\right)}^3=0$

${\mathrm{ode}}_2≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-2{y\left(x\right)}^3=0$

$\mathrm{gensol}\left[2\right]≔\mathrm{dsolve}\left(\mathrm{ode}\left[2\right]\right)$

${\mathrm{gensol}}_2≔y\left(x\right)=\mathrm{c\_\_2}\mathrm{JacobiSN}\left(\left(\mathrm{I}x+\mathrm{c\_\_1}\right)\mathrm{c\_\_2}\,\mathrm{I}\right)$

$\mathrm{ic}\left[2\right]≔y\left(0\right)=1,\mathrm{D}\left(y\right)\left(0\right)=1$

${\mathrm{ic}}_2≔y\left(0\right)=1,\mathrm{D}\left(y\right)\left(0\right)=1$

$\mathrm{dsolve}\left(\left[\mathrm{ode}\left[2\right]\,\mathrm{ic}\left[2\right]\right]\right)$

$y\left(x\right)=-\frac{1}{-1+x}$

$\mathrm{DEtools}\left[\mathrm{IVPsol}\right]\left(\left[\mathrm{ic}\left[2\right]\right]\,\mathrm{gensol}\left[2\right]\right)$

$y\left(x\right)=\mathrm{RootOf}\left(\mathrm{I}{\mathrm{\_Z}}^2\mathrm{JacobiCN}\left(\mathrm{InverseJacobiSN}\left(\frac{1}{\mathrm{\_Z}}\,\mathrm{I}\right)\,\mathrm{I}\right)\mathrm{JacobiDN}\left(\mathrm{InverseJacobiSN}\left(\frac{1}{\mathrm{\_Z}}\,\mathrm{I}\right)\,\mathrm{I}\right)-1\right)\mathrm{JacobiSN}\left(\left(\mathrm{I}x+\frac{\mathrm{InverseJacobiSN}\left(\frac{1}{\mathrm{RootOf}\left(\mathrm{I}{\mathrm{\_Z}}^2\mathrm{JacobiCN}\left(\mathrm{InverseJacobiSN}\left(\frac{1}{\mathrm{\_Z}}\,\mathrm{I}\right)\,\mathrm{I}\right)\mathrm{JacobiDN}\left(\mathrm{InverseJacobiSN}\left(\frac{1}{\mathrm{\_Z}}\,\mathrm{I}\right)\,\mathrm{I}\right)-1\right)}\,\mathrm{I}\right)}{\mathrm{RootOf}\left(\mathrm{I}{\mathrm{\_Z}}^2\mathrm{JacobiCN}\left(\mathrm{InverseJacobiSN}\left(\frac{1}{\mathrm{\_Z}}\,\mathrm{I}\right)\,\mathrm{I}\right)\mathrm{JacobiDN}\left(\mathrm{InverseJacobiSN}\left(\frac{1}{\mathrm{\_Z}}\,\mathrm{I}\right)\,\mathrm{I}\right)-1\right)}\right)\mathrm{RootOf}\left(\mathrm{I}{\mathrm{\_Z}}^2\mathrm{JacobiCN}\left(\mathrm{InverseJacobiSN}\left(\frac{1}{\mathrm{\_Z}}\,\mathrm{I}\right)\,\mathrm{I}\right)\mathrm{JacobiDN}\left(\mathrm{InverseJacobiSN}\left(\frac{1}{\mathrm{\_Z}}\,\mathrm{I}\right)\,\mathrm{I}\right)-1\right)\,\mathrm{I}\right)$

$\mathrm{gensol}\left[2.1\right]≔\mathrm{dsolve}\left(\mathrm{ode}\left[2\right]\,\mathrm{Lie}\,\mathrm{explicit}\right)$

${\mathrm{gensol}}_{2.1}≔y\left(x\right)=\mathrm{RootOf}\left(-\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{\sqrt{{\mathrm{\_f}}^4+\mathrm{c\_\_1}}}\ⅆ\mathrm{\_f}\right)+x+\mathrm{c\_\_2}\right),y\left(x\right)=\mathrm{RootOf}\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{\sqrt{{\mathrm{\_f}}^4+\mathrm{c\_\_1}}}\ⅆ\mathrm{\_f}+x+\mathrm{c\_\_2}\right)$

$\mathrm{DEtools}\left[\mathrm{IVPsol}\right]\left(\left[\mathrm{ic}\left[2\right]\right]\,\mathrm{gensol}\left[2.1\right]\left[1\right]\right)$

$y\left(x\right)=\mathrm{RootOf}\left({\int }_{\mathrm{\_Z}}^1\frac{1}{\sqrt{{\mathrm{\_f}}^4}}\ⅆ\mathrm{\_f}+x\right)$

$\mathrm{partsol}\left[2\right]≔\mathrm{DEtools}\left[\mathrm{particularsol}\right]\left(\mathrm{ode}\left[2\right]\right)\left[2\right]$

${\mathrm{partsol}}_2≔y\left(x\right)=\frac{1}{-x+\mathrm{c\_\_1}}$

$\mathrm{DEtools}\left[\mathrm{IVPsol}\right]\left(\left[\mathrm{ic}\left[2\right]\right]\,\mathrm{partsol}\left[2\right]\right)$

$y\left(x\right)=\frac{1}{1-x}$