Integrate - Maple Help

Student[ODEs]

 Integrate
 integrate an exact ODE

 Calling Sequence Integrate(ODE, y(x)) Integrate:-Apply(ODE, y(x)) Integrate:-Evaluate(eq)

Parameters

 ODE - a first order ordinary differential equation y - name; the dependent variable x - name; the independent variable

Description

 • Integrate(ODE, y(x)) applies an integral to both sides of an exact ODE, and then performs the integration, thereby obtaining a solution to the ODE.
 • Integrate:-Apply(ODE, y(x)) applies an integral to both sides of an exact ODE, without performing the integration.
 • Integrate:-Evaluate(ODE) tries to evaluate any unevaluated integrals in the expression ODE, if possible.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\right):$
 > $\mathrm{ode1}≔\frac{\mathrm{diff}\left(z\left(t\right),t\right)}{z\left(t\right)+1}{z\left(t\right)}^{2}=-\frac{{t}^{2}}{t-1}$
 ${\mathrm{ode1}}{≔}\frac{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({t}\right)\right){}{{z}{}\left({t}\right)}^{{2}}}{{z}{}\left({t}\right){+}{1}}{=}{-}\frac{{{t}}^{{2}}}{{t}{-}{1}}$ (1)
 > $\mathrm{Integrate}\left(\mathrm{ode1},z\left(t\right)\right)$
 $\frac{{{z}{}\left({t}\right)}^{{2}}}{{2}}{-}{z}{}\left({t}\right){+}{\mathrm{ln}}{}\left({z}{}\left({t}\right){+}{1}\right){=}{-}\frac{{{t}}^{{2}}}{{2}}{-}{t}{-}{\mathrm{ln}}{}\left({t}{-}{1}\right){+}{\mathrm{_C1}}$ (2)
 > $\mathrm{eq2}≔\mathrm{Integrate}:-\mathrm{Apply}\left(\mathrm{ode1},z\left(t\right)\right)$
 ${\mathrm{eq2}}{≔}{\int }\frac{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({t}\right)\right){}{{z}{}\left({t}\right)}^{{2}}}{{z}{}\left({t}\right){+}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}{=}{\int }{-}\frac{{{t}}^{{2}}}{{t}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}{+}{\mathrm{_C1}}$ (3)
 > $\mathrm{Integrate}:-\mathrm{Evaluate}\left(\mathrm{eq2}\right)$
 $\frac{{{z}{}\left({t}\right)}^{{2}}}{{2}}{-}{z}{}\left({t}\right){+}{\mathrm{ln}}{}\left({z}{}\left({t}\right){+}{1}\right){=}{-}\frac{{{t}}^{{2}}}{{2}}{-}{t}{-}{\mathrm{ln}}{}\left({t}{-}{1}\right){+}{\mathrm{_C1}}$ (4)

Compatibility

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