The odetest command checks explicit and implicit solutions for ODEs by making a careful simplification of the ODE with respect to the given solution. If the solution is valid, the returned result will be 0; otherwise, the algebraic remaining expression will be returned. In the case of systems of ODEs, odetest can only test explicit solutions, given either as a set or as a list of sets. (For information on non-composed sets of solutions for nonlinear systems, see dsolve,system.)

To test whether a solution satisfies one or many initial or boundary conditions, pass to odetest the ODE together with the initial or boundary conditions, enclosed as a set or list, as second argument.

If odetest returns a nonzero result, the solution being tested is not necessarily wrong; sometimes further simplifications or manipulations of odetest's output are required to obtain zero, and so verify the solution is correct. If the solution was obtained using the dsolve command, it is recommended that you recompute the solution using one or both of the useInt and implicit options - see dsolve. This may facilitate the verification process. Also, an alternative testing technique, particularly useful with linear ODEs, is to try to recompute the ODE departing from the solution which odetest fails in testing. Examples of both types are found at the end of the next section.

To test series solutions, pass the keyword series as an extra argument. Only one series solution for one ODE (can be a set with initial/boundary conditions) can be tested.

$\mathrm{ODE}≔\left[\mathrm{diff}\left(y\left(x\right)\,x\right)=\sin \left(x-y\left(x\right)\right)\,y\left(0\right)=8\right]$

$\mathrm{ODE}≔\left[\frac{\ⅆ}{\ⅆx}y\left(x\right)=\sin \left(x-y\left(x\right)\right)\,y\left(0\right)=8\right]$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ODE}\right)$

$\mathrm{sol}≔y\left(x\right)=x-2\arctan \left(\frac{\tan \left(4\right)x-2\tan \left(4\right)+x}{\tan \left(4\right)x+x+2}\right)+2\mathrm{\pi }$

$\mathrm{odetest}\left(\mathrm{sol}\,\mathrm{ODE}\right)$

$\left[0\,0\right]$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\mathrm{diff}\left(y\left(x\right)\,x\right)+y\left(x\right)=0$

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

$\mathrm{bc}≔y\left(0\right)=1,y\left(2\mathrm{\pi }\right)=1$

$\mathrm{bc}≔y\left(0\right)=1,y\left(2\mathrm{\pi }\right)=1$

$\mathrm{sol}≔\mathrm{dsolve}\left(\left\{\mathrm{ODE}\,\mathrm{bc}\right\}\right)$

$\mathrm{sol}≔y\left(x\right)={\ⅇ}^{-\frac{x}{2}}\left(\left(\csc \left(\sqrt{3}\mathrm{\pi }\right){\ⅇ}^{\mathrm{\pi }}-\cot \left(\sqrt{3}\mathrm{\pi }\right)\right)\sin \left(\frac{\sqrt{3}x}{2}\right)+\cos \left(\frac{\sqrt{3}x}{2}\right)\right)$

$\mathrm{odetest}\left(\mathrm{sol}\,\left[\mathrm{ODE}\,\mathrm{bc}\right]\right)$

$\left[0\,0\,0\right]$

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

$\mathrm{ODE}≔\left[\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2=0\,y\left(a\right)=0\,\mathrm{D}\left(y\right)\left(a\right)=1\right]$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ODE}\,y\left(x\right)\,\mathrm{type}='\mathrm{series}'\right)$

$\mathrm{sol}≔y\left(x\right)=\left(x-a\right)-\frac{1}{2}{\left(x-a\right)}^2+\frac{1}{3}{\left(x-a\right)}^3-\frac{1}{4}{\left(x-a\right)}^4+\frac{1}{5}{\left(x-a\right)}^5+O\left({\left(x-a\right)}^6\right)$

$\mathrm{odetest}\left(\mathrm{sol}\,\mathrm{ODE}\,\mathrm{series}\right)$

$\left[0\,0\,0\right]$

$\mathrm{ODE}≔\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)=\left(3x^2+c\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+\left(\left(3-b\right)x-a\right)y\left(x\right)$

$\mathrm{ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\left(3x^2+c\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(\left(3-b\right)x-a\right)y\left(x\right)$

$\mathrm{sol}≔y\left(x\right)=\mathrm{series}\left(1+\left(-\frac{1}{2}a\right)x^2+\left(-\frac{1}{6}b+\frac{1}{2}-\frac{1}{6}ca\right)x^3+\left(\frac{1}{24}{a}^2-\frac{1}{24}cb+\frac{1}{8}c-\frac{1}{24}{c}^{2}a\right)x^4+\mathrm{O}\left(x^5\right)\,x\,5\right)$

$\mathrm{sol}≔y\left(x\right)=1-\frac{1}{2}a{x}^2+\left(-\frac{b}{6}+\frac{1}{2}-\frac{ca}{6}\right)x^3+\left(\frac{1}{24}{a}^2-\frac{1}{24}cb+\frac{1}{8}c-\frac{1}{24}{c}^{2}a\right)x^4+O\left(x^5\right)$

$\mathrm{odetest}\left(\mathrm{sol}\,\mathrm{ODE}\,\mathrm{series}\,\mathrm{point}=0\right)$

$0$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=\frac{1}{x^2}\mathrm{\_F1}\left(\frac{\mathrm{diff}\left(y\left(x\right)\,x\right)x}{y\left(x\right)}\right)y\left(x\right)$

$\mathrm{ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{\mathrm{\_F1}\left(\frac{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x}{y\left(x\right)}\right)y\left(x\right)}{x^2}$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ODE}\right)$

$\mathrm{sol}≔y\left(x\right)={\ⅇ}^{{\int }_{\phantom{\mathrm{`\; `}}}^{\ln \left(x\right)}\mathrm{RootOf}\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{\mathrm{\_a}-{\mathrm{\_a}}^2+\mathrm{\_F1}\left(\mathrm{\_a}\right)}\ⅆ\mathrm{\_a}-\mathrm{\_b}+\mathrm{c\_\_1}\right)\ⅆ\mathrm{\_b}+\mathrm{c\_\_2}}$

$\mathrm{odetest}\left(\mathrm{sol}\,\mathrm{ODE}\right)$

$0$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=F\left(\frac{y\left(x\right)-x\ln \left(x\right)}{x}\right)+\ln \left(x\right)$

$\mathrm{ODE}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=F\left(\frac{y\left(x\right)-x\ln \left(x\right)}{x}\right)+\ln \left(x\right)$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ODE}\,\mathrm{implicit}\right)$

$\mathrm{sol}≔\ln \left(x\right)-\left({\int }_{\phantom{\mathrm{`\; `}}}^{-\ln \left(x\right)+\frac{y\left(x\right)}{x}}\frac{1}{-1+F\left(\mathrm{\_a}\right)-\mathrm{\_a}}\ⅆ\mathrm{\_a}\right)-\mathrm{c\_\_1}=0$

$\mathrm{odetest}\left(\mathrm{sol}\,\mathrm{ODE}\right)$

$0$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=x{f\left(x\right)}^2\left(x+2\right){y\left(x\right)}^3+f\left(x\right)\left(x+3\right){y\left(x\right)}^2-\frac{\mathrm{diff}\left(f\left(x\right)\,x\right)}{f\left(x\right)}y\left(x\right)$

$\mathrm{ODE}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=x{f\left(x\right)}^2\left(x+2\right){y\left(x\right)}^3+f\left(x\right)\left(x+3\right){y\left(x\right)}^2-\frac{\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)y\left(x\right)}{f\left(x\right)}$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ODE}\,y\left(x\right)\,\mathrm{implicit}\right)$

$\mathrm{sol}≔\mathrm{c\_\_1}+\mathrm{arctanh}\left(\frac{\sqrt{f\left(x\right)y\left(x\right)}x}{\sqrt{xf\left(x\right)y\left(x\right)\left(x+2\right)+2}}\right)+\frac{\sqrt{xf\left(x\right)y\left(x\right)\left(x+2\right)+2}}{2\sqrt{f\left(x\right)y\left(x\right)}}=0$

$\mathrm{odetest}\left(\mathrm{sol}\,\mathrm{ODE}\,y\left(x\right)\right)$

$0$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=\frac{\mathrm{diff}\left(y\left(x\right)\,x\right)-{y\left(x\right)}^3-f\left(x\right)+3x{y\left(x\right)}^2\mathrm{diff}\left(y\left(x\right)\,x\right)+x\mathrm{diff}\left(f\left(x\right)\,x\right)}{x}$

$\mathrm{ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)-{y\left(x\right)}^3-f\left(x\right)+3x{y\left(x\right)}^2\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)x}{x}$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ODE}\,y\left(x\right)\right)$

$\mathrm{sol}≔y\left(x\right)=\mathrm{\_b}\left(\mathrm{\_a}\right)\mathbf{where}\left[\left\{\frac{\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)}{\mathrm{\_a}}-\frac{{\mathrm{\_b}\left(\mathrm{\_a}\right)}^3-\mathrm{c\_\_1}\mathrm{\_a}+f\left(\mathrm{\_a}\right)}{\mathrm{\_a}}=0\right\}\,\left\{\mathrm{\_a}=x\,\mathrm{\_b}\left(\mathrm{\_a}\right)=y\left(x\right)\right\}\,\left\{x=\mathrm{\_a}\,y\left(x\right)=\mathrm{\_b}\left(\mathrm{\_a}\right)\right\}\right]$

$\mathrm{odetest}\left(\mathrm{sol}\,\mathrm{ODE}\right)$

$0$

$\mathrm{sysODE}≔\left\{\mathrm{diff}\left(x\left(t\right)\,t\right)=y\left(t\right)\,\mathrm{diff}\left(y\left(t\right)\,t\right)=-x\left(t\right)\right\},\left\{x\,y\right\}\left(t\right)$

$\mathrm{sysODE}≔\left\{\frac{\ⅆ}{\ⅆt}x\left(t\right)=y\left(t\right)\,\frac{\ⅆ}{\ⅆt}y\left(t\right)=-x\left(t\right)\right\},\left\{x\left(t\right)\,y\left(t\right)\right\}$

$\mathrm{solsys}≔\mathrm{dsolve}\left(\mathrm{sysODE}\right)$

$\mathrm{solsys}≔\left\{x\left(t\right)=\mathrm{c\_\_1}\sin \left(t\right)+\mathrm{c\_\_2}\cos \left(t\right)\,y\left(t\right)=\mathrm{c\_\_1}\cos \left(t\right)-\mathrm{c\_\_2}\sin \left(t\right)\right\}$

$\mathrm{odetest}\left(\mathrm{solsys}\,\mathrm{sysODE}\right)$

$\left\{0\right\}$

$\mathrm{sysODE}≔\left\{\mathrm{diff}\left(x\left(t\right)\,t\right)=y\left(t\right)\,\mathrm{diff}\left(y\left(t\right)\,t\right)=-{x\left(t\right)}^2\right\},\left\{x\,y\right\}\left(t\right)$

$\mathrm{sysODE}≔\left\{\frac{\ⅆ}{\ⅆt}x\left(t\right)=y\left(t\right)\,\frac{\ⅆ}{\ⅆt}y\left(t\right)=-{x\left(t\right)}^2\right\},\left\{x\left(t\right)\,y\left(t\right)\right\}$

$\mathrm{solsys}≔\mathrm{dsolve}\left(\mathrm{sysODE}\right)$

$\mathrm{solsys}≔\left[\left\{x\left(t\right)=-6\mathrm{WeierstrassP}\left(t+\mathrm{c\_\_1}\,0\,\mathrm{c\_\_2}\right)\right\}\,\left\{y\left(t\right)=\frac{\ⅆ}{\ⅆt}x\left(t\right)\right\}\right]$

$\mathrm{odetest}\left(\mathrm{solsys}\,\mathrm{sysODE}\right)$

$\left\{0\right\}$

$\mathrm{solsys}≔\mathrm{dsolve}\left(\mathrm{sysODE}\,\mathrm{explicit}\right)$

$\mathrm{solsys}≔\left\{x\left(t\right)=-6\mathrm{WeierstrassP}\left(t+\mathrm{c\_\_1}\,0\,\mathrm{c\_\_2}\right)\,y\left(t\right)=-6\mathrm{WeierstrassPPrime}\left(t+\mathrm{c\_\_1}\,0\,\mathrm{c\_\_2}\right)\right\}$

$\mathrm{map}\left(\mathrm{odetest}\,\left[\mathrm{solsys}\right]\,\mathrm{sysODE}\right)$

$\left[\left\{0\right\}\right]$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(t\right)\,t\right)=\frac{\left(b+2+2t\right)y\left(t\right)+1}{1-{\left(1+t\right)}^2}$

$\mathrm{ODE}≔\frac{\ⅆ}{\ⅆt}y\left(t\right)=\frac{\left(b+2+2t\right)y\left(t\right)+1}{1-{\left(1+t\right)}^2}$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ODE}\right)$

$\mathrm{sol}≔y\left(t\right)=\left(-\frac{2^{2-\frac{b}{2}}{t}^{\frac{b}{2}+1}\mathrm{hypergeom}\left(\left[\frac{b}{2}\,\frac{b}{2}+1\right]\,\left[2+\frac{b}{2}\right]\,-\frac{t}{2}\right)}{2\left(b+2\right)}+\mathrm{c\_\_1}\right){\left(t+2\right)}^{\frac{b}{2}-1}{t}^{-\frac{b}{2}-1}$

$\mathrm{odetest}\left(\mathrm{sol}\,\mathrm{ODE}\right)$

$-\frac{\left(-{\left(1+\frac{t}{2}\right)}^{\frac{b}{2}}\mathrm{hypergeom}\left(\left[\frac{b}{2}+1\,\frac{b}{2}+1\right]\,\left[2+\frac{b}{2}\right]\,-\frac{t}{2}\right)bt+2\mathrm{hypergeom}\left(\left[\frac{b}{2}\,\frac{b}{2}\right]\,\left[\frac{b}{2}+1\right]\,-\frac{t}{2}\right)b{\left(1+\frac{t}{2}\right)}^{\frac{b}{2}}+4{\left(1+\frac{t}{2}\right)}^{\frac{b}{2}}\mathrm{hypergeom}\left(\left[\frac{b}{2}\,\frac{b}{2}\right]\,\left[\frac{b}{2}+1\right]\,-\frac{t}{2}\right)-2b-4\right)\left(b+t\right)\left(b+2+t\right)}{2{\left(t+2\right)}^2{t}^2\left(b+2\right)}$

$\mathrm{sol2}≔\mathrm{dsolve}\left(\mathrm{ODE}\,\mathrm{useInt}\right)$

$\mathrm{sol2}≔y\left(t\right)=\left(\int -\frac{{\ⅇ}^{\int \frac{b+2+2t}{t\left(t+2\right)}\ⅆt}}{t\left(t+2\right)}\ⅆt+\mathrm{c\_\_1}\right){\ⅇ}^{\int -\frac{b+2+2t}{t\left(t+2\right)}\ⅆt}$

$\mathrm{odetest}\left(\mathrm{sol2}\,\mathrm{ODE}\right)$

$0$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)=\frac{\left(\left(-a\left[1\right]F+a\left[0\right]E\right)x+Bea\left[1\right]\right)y\left(x\right)}{B^2{e}^{2}x{E}^3}$

$\mathrm{ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{\left(\left(a_{0}E-a_{1}F\right)x+Be{a}_1\right)y\left(x\right)}{B^2{e}^{2}x{E}^3}$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ODE}\,\left[\mathrm{hyper3}\right]\right)$

$\mathrm{sol}≔y\left(x\right)=\mathrm{c\_\_1}{\ⅇ}^{-\frac{\sqrt{a_{0}E-a_{1}F}x}{Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}x\mathrm{hypergeom}\left(\left[\frac{2\sqrt{a_{0}E-a_{1}F}{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+a_1}{2\sqrt{a_{0}E-a_{1}F}{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right]\,\left[2\right]\,\frac{2\sqrt{a_{0}E-a_{1}F}x}{Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right)+\mathrm{c\_\_2}{\ⅇ}^{-\frac{\sqrt{a_{0}E-a_{1}F}x}{Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}x\mathrm{KummerU}\left(\frac{2\sqrt{a_{0}E-a_{1}F}{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+a_1}{2\sqrt{a_{0}E-a_{1}F}{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,2\,\frac{2\sqrt{a_{0}E-a_{1}F}x}{Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right)$

$\mathrm{ode}≔\mathrm{PDEtools}\left[\mathrm{dpolyform}\right]\left(\mathrm{sol}\,\mathrm{no\_Fn}\right)$

$\mathrm{ode}≔\left[\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{\left(Be{a}_1+Ex{a}_0-Fx{a}_1\right)y\left(x\right)}{B^2{e}^{2}x{E}^3}\right]\mathbf{where}\left[\right]$

$\mathrm{normal}\left(\mathrm{ODE}-\mathrm{op}\left(\left[1\,1\right]\,\mathrm{ode}\right)\right)$

$0=0$

$\mathrm{sol\_W}≔\mathrm{convert}\left(\mathrm{sol}\,\mathrm{Whittaker}\right)$

$\mathrm{sol\_W}≔y\left(x\right)=-\frac{\mathrm{c\_\_1}\mathrm{WhittakerM}\left(\mathrm{-1}\,\frac{1}{2}\,-\frac{\sqrt{a_{0}E-a_{1}F}x}{Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right)B^2{e}^2{E}^3\mathrm{WhittakerM}\left(-\frac{a_1}{2\sqrt{a_{0}E-a_{1}F}{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\frac{1}{2}\,\frac{2\sqrt{a_{0}E-a_{1}F}x}{Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right){\ⅇ}^{\frac{\sqrt{a_{0}E-a_{1}F}x}{Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}}{2{\ⅇ}^{\frac{\sqrt{a_{0}E-a_{1}F}x}{2Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(a_{0}E-a_{1}F\right)x}-\frac{\mathrm{c\_\_2}\mathrm{WhittakerM}\left(\mathrm{-1}\,\frac{1}{2}\,-\frac{\sqrt{a_{0}E-a_{1}F}x}{Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right)B^2{e}^2{E}^3\mathrm{WhittakerW}\left(-\frac{a_1}{2\sqrt{a_{0}E-a_{1}F}{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\,\frac{1}{2}\,\frac{2\sqrt{a_{0}E-a_{1}F}x}{Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right){\ⅇ}^{\frac{\sqrt{a_{0}E-a_{1}F}x}{Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}}{2{\ⅇ}^{\frac{\sqrt{a_{0}E-a_{1}F}x}{2Be{E}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left(a_{0}E-a_{1}F\right)x}$

$\mathrm{odetest}\left(\mathrm{sol\_W}\,\mathrm{ODE}\right)$

$0$