The Test(sol, ODE, y(x)) tests whether the candidate solution sol is in fact a valid solution of the given equation or system ODE.

The equation or system ODE is evaluated using the supplied candidate solution sol, and an attempt is made to remove all occurrences of the dependent variables var and their derivatives; whatever remains is then simplified and returned.

If this remainder is 0 (or a set or list of zeros), then sol is in fact a valid solution.

Conversely, if sol is in fact a valid solution, then this remainder must be mathematically equivalent to 0 (or a set or list of zeros), but it may not have been simplified fully.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\right)\:$

$\mathrm{ode1}≔t^2\left(z\left(t\right)+1\right)+{z\left(t\right)}^2\left(t-1\right)\mathrm{diff}\left(z\left(t\right)\,t\right)=0$

$\mathrm{ode1}≔t^2\left(z\left(t\right)+1\right)+{z\left(t\right)}^2\left(t-1\right)\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)=0$

$\mathrm{sol1}≔\mathrm{Solve}\left(\mathrm{ode1}\,z\left(t\right)\right)$

$\mathrm{sol1}≔\frac{{z\left(t\right)}^2}{2}-z\left(t\right)+\ln \left(z\left(t\right)+1\right)=-\frac{t^2}{2}-t-\ln \left(t-1\right)+\mathrm{c\_\_1}$

$\mathrm{Test}\left(\mathrm{sol1}\,\mathrm{ode1}\,z\left(t\right)\right)$

$0$

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-\mathrm{diff}\left(y\left(x\right)\,x\right)-x\exp \left(x\right)=0$

$\mathrm{ode2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-\frac{\ⅆ}{\ⅆx}y\left(x\right)-x{\ⅇ}^x=0$

$\mathrm{sol2}≔\mathrm{Solve}\left(\mathrm{ode2}\,y\left(x\right)\right)$

$\mathrm{sol2}≔y\left(x\right)=\mathrm{c\_\_1}+\mathrm{c\_\_2}{\ⅇ}^x+{\ⅇ}^x\left(1-x+\frac{1}{2}{x}^2\right)$

$\mathrm{Test}\left(\mathrm{sol2}\,\mathrm{ode2}\,y\left(x\right)\right)$

$0$

$\mathrm{ode3}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\frac{5{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2}{y\left(x\right)}=0$

$\mathrm{ode3}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\frac{5{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}{y\left(x\right)}=0$

$\mathrm{sol3}≔\mathrm{Solve}\left(\mathrm{ode3}\,y\left(x\right)\right)$

$\mathrm{sol3}≔\left\{y\left(x\right)={\left(\mathrm{c\_\_1}x+\mathrm{c\_\_2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\,y\left(x\right)=\mathrm{c\_\_1}{\left(x+\mathrm{c\_\_2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\,y\left(x\right)=\mathrm{c\_\_2}{\left(\mathrm{c\_\_1}x+1\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right\}$

$\mathrm{Test}\left(\mathrm{sol3}\,\mathrm{ode3}\,y\left(x\right)\right)$

$\left\{0\right\}$

$\mathrm{ode4}≔x^3\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+3x^2\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-6x\mathrm{diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)=0$

$\mathrm{ode4}≔x^3\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+3x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-6x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-6y\left(x\right)=0$

$\mathrm{sol4}≔\mathrm{Solve}\left(\mathrm{ode4}\,y\left(x\right)\right)$

$\mathrm{sol4}≔y\left(x\right)=\frac{\mathrm{c\_\_1}{x}^5+\mathrm{c\_\_3}x+\mathrm{c\_\_2}}{x^2}$

$\mathrm{Test}\left(\mathrm{sol4}\,\mathrm{ode4}\,y\left(x\right)\right)$

$0$

$\mathrm{ode5}≔\left\{\mathrm{diff}\left(y\left[1\right]\left(x\right)\,x\right)=7y\left[1\right]\left(x\right)+y\left[2\right]\left(x\right)\,\mathrm{diff}\left(y\left[2\right]\left(x\right)\,x\right)=-4y\left[1\right]\left(x\right)+3y\left[2\right]\left(x\right)\right\}$

$\mathrm{ode5}≔\left\{\frac{\ⅆ}{\ⅆx}{y}_1\left(x\right)=7y_1\left(x\right)+y_2\left(x\right)\,\frac{\ⅆ}{\ⅆx}{y}_2\left(x\right)=-4y_1\left(x\right)+3y_2\left(x\right)\right\}$

$\mathrm{sol5}≔\mathrm{Solve}:-\mathrm{System}\left(\mathrm{ode5}\,\mathrm{output}=\mathrm{solution}\right)$

$\mathrm{sol5}≔\left\{y_1\left(x\right)=-\frac{{\ⅇ}^{5x}\left(2\mathrm{c\_\_2}x+2\mathrm{c\_\_1}+\mathrm{c\_\_2}\right)}{4}\,y_2\left(x\right)={\ⅇ}^{5x}\left(\mathrm{c\_\_2}x+\mathrm{c\_\_1}\right)\right\}$

$\mathrm{Test}\left(\mathrm{sol5}\,\mathrm{ode5}\,\left\{\mathrm{seq}\left(y\left[j\right]\left(x\right)\,j=1..2\right)\right\}\right)$

$\left\{0\right\}$

The Student[ODEs][Test] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.