DESol is a data structure to represent the solution of a differential equation. It is to dsolve as RootOf is to solve.

The DESol data structure makes manipulation of differential equations much easier. Its existence allows many hooks to the rest of Maple, diff, int, evalf, series, simplify, just to name the most prominent.

DESol can be used as an operator (using D), or as an expression (using diff(...,x)). The examples below illustrate some of the operations that can be performed with DESol.

$\mathrm{de1}≔\mathrm{DESol}\left(\mathrm{D}\left(y\right)-y\,y\right)$

$\mathrm{de1}≔\mathrm{DESol}\left(\left\{\mathrm{D}\left(y\right)-y\right\}\,\left\{y\right\}\right)$

$\mathrm{D}\left(\mathrm{de1}\right)-\mathrm{de1}$

$0$

$\mathrm{de2}≔\mathrm{DESol}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)-y\left(x\right)\,y\left(x\right)\right)$

$\mathrm{de2}≔\mathrm{DESol}\left(\left\{\frac{\ⅆ}{\ⅆx}y\left(x\right)-y\left(x\right)\right\}\,\left\{y\left(x\right)\right\}\right)$

$\mathrm{de1}\left(x\right)-\mathrm{de2}$

$0$

$\mathrm{diff}\left(\mathrm{de2}\,x\right)-\mathrm{de2}$

$0$

$\mathrm{unapply}\left(\mathrm{de2}\,x\right)-\mathrm{de1}$

$0$

$\mathrm{alias}\left(\mathrm{de3}=\mathrm{DESol}\left(\mathrm{D}\left(\mathrm{D}\left(y\right)\right)-ax\mathrm{D}\left(y\right)+y\,y\right)\right)\:$$\mathrm{D}\left(x\right)≔1\:$$\mathrm{D}\left(a\right)≔0\:$

$\mathrm{D}\left(\mathrm{D}\left(\mathrm{D}\left(\mathrm{de3}\right)\right)\right)$

$a\mathrm{D}\left(\mathrm{de3}\right)+ax\left(ax\mathrm{D}\left(\mathrm{de3}\right)-\mathrm{de3}\right)-\mathrm{D}\left(\mathrm{de3}\right)$

$\mathrm{collect}\left(\mathrm{D}\left(\mathrm{D}\left(\mathrm{D}\left(\mathrm{D}\left(\mathrm{de3}\right)\right)\right)\right)\,\mathrm{D}\left(\mathrm{de3}\right)\,\mathrm{factor}\right)$

$ax\left(a^2{x}^2+3a-2\right)\mathrm{D}\left(\mathrm{de3}\right)-\mathrm{de3}\left(a^2{x}^2+2a-1\right)$

$\mathrm{int}\left(\mathrm{de3}\left(x\right)\,x\right)$

$\mathrm{DESol}\left(\left\{y\left(x\right)-\frac{xa\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{a+1}+\frac{\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)}{a+1}\right\}\,\left\{y\left(x\right)\right\}\right)$

$\mathrm{int}\left(\frac{1}{\ln \left(\mathrm{de3}\left(x\right)\right)}\,x\right)$

$\mathrm{DESol}\left(\left\{\frac{2{\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}^2{\ⅇ}^{\frac{1}{\frac{\ⅆ}{\ⅆx}y\left(x\right)}}}{{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^3}-\frac{\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right){\ⅇ}^{\frac{1}{\frac{\ⅆ}{\ⅆx}y\left(x\right)}}}{{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}+\frac{{\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}^2{\ⅇ}^{\frac{1}{\frac{\ⅆ}{\ⅆx}y\left(x\right)}}}{{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^4}+\frac{ax\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right){\ⅇ}^{\frac{1}{\frac{\ⅆ}{\ⅆx}y\left(x\right)}}}{{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}+{\ⅇ}^{\frac{1}{\frac{\ⅆ}{\ⅆx}y\left(x\right)}}\right\}\,\left\{y\left(x\right)\right\}\right)$

$\mathrm{alias}\left(\mathrm{de5}=\mathrm{DESol}\left(\frac{\mathrm{D}\left(\mathrm{D}\left(y\right)\right)}{\mathrm{D}\left(y\right)}=y\right)\right)\:$

$\frac{\mathrm{D}\left(\mathrm{D}\left(\mathrm{de5}\right)\right)}{\mathrm{D}\left(\mathrm{de5}\right)}-\mathrm{de5}$

$0$

$\mathrm{D}\left(\mathrm{D}\left(\mathrm{D}\left(\mathrm{de5}\right)\right)\right)$

${\mathrm{D}\left(\mathrm{de5}\right)}^2+{\mathrm{de5}}^2\mathrm{D}\left(\mathrm{de5}\right)$

$\mathrm{D}\left(x\right)≔'\mathrm{D}\left(x\right)'\:$

$\mathrm{de6}≔\mathrm{DESol}\left(\mathrm{diff}\left(x\left(t\right)\,t\,t\right)=\frac{g}{l}\sin \left(x\left(t\right)\right)\,x\left(t\right)\,\left\{x\left(0\right)=0\,\mathrm{D}\left(x\right)\left(0\right)=\mathrm{v0}\right\}\right)$

$\mathrm{de6}≔\mathrm{DESol}\left(\left\{\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)-\frac{g\sin \left(x\left(t\right)\right)}{l}\right\}\,\left\{x\left(t\right)\right\}\,\left\{x\left(0\right)=0\,\mathrm{D}\left(x\right)\left(0\right)=\mathrm{v0}\right\}\right)$

$\mathrm{de7}≔\mathrm{int}\left(\mathrm{de6}\,t\right)$

$\mathrm{de7}≔\mathrm{DESol}\left(\left\{\frac{{\ⅆ}^3}{\ⅆt^3}x\left(t\right)-\frac{g\sin \left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}{l}\right\}\,\left\{x\left(t\right)\right\}\right)$

$\mathrm{diff}\left(\mathrm{de7}\,t\,t\,t\right)-\frac{g}{l}\sin \left(\mathrm{diff}\left(\mathrm{de7}\,t\right)\right)$

$0$

$\mathrm{de8}≔\mathrm{int}\left(\frac{1}{\mathrm{DESol}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)-y\left(x\right)\,y\left(x\right)\right)+1}\,x\right)$

$\mathrm{de8}≔-\ln \left(\mathrm{DESol}\left(\left\{\frac{\ⅆ}{\ⅆx}y\left(x\right)-y\left(x\right)\right\}\,\left\{y\left(x\right)\right\}\right)+1\right)+\ln \left(\mathrm{DESol}\left(\left\{\frac{\ⅆ}{\ⅆx}y\left(x\right)-y\left(x\right)\right\}\,\left\{y\left(x\right)\right\}\right)\right)$

$\mathrm{normal}\left(\mathrm{diff}\left(\mathrm{de8}\,x\right)-\frac{1}{\mathrm{DESol}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)-y\left(x\right)\,y\left(x\right)\right)+1}\right)$

$0$

$\mathrm{DESol}\left({y\left(x\right)}^2-y\left(x\right)+1\,y\left(x\right)\right)$

$\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}+1\right)$

$\mathrm{DESol}\left(y\left(x\right)-x\,y\left(x\right)\right)$

$x$

$\mathrm{de9}≔\mathrm{DESol}\left(\mathrm{diff}\left(y\left(x\right)\,x\,x\right)\mathrm{diff}\left(y\left(x\right)\,x\right)-1\,y\left(x\right)\right)$

$\mathrm{de9}≔\mathrm{DESol}\left(\left\{\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-1\right\}\,\left\{y\left(x\right)\right\}\right)$

$\mathrm{diff}\left(\mathrm{de9}\,x\right)$

$\mathrm{DESol}\left(\left\{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)y\left(x\right)-1\right\}\,\left\{y\left(x\right)\right\}\right)$

$\mathrm{de10}≔\mathrm{unapply}\left(\mathrm{de9}\,x\right)$

$\mathrm{de10}≔\mathrm{DESol}\left(\left\{{\mathrm{D}}^{\left(2\right)}\left(y\right)\mathrm{D}\left(y\right)-1\right\}\,\left\{y\right\}\right)$

$\mathrm{D}\left(\mathrm{de10}\right)$

$\mathrm{DESol}\left(\left\{y\mathrm{D}\left(y\right)-1\right\}\,\left\{y\right\}\right)$