ByLaplaceTransform - Maple Help

Student[ODEs][Solve]

 ByLaplaceTransform
 Solve a linear ODE using the Laplace transform

 Calling Sequence ByLaplaceTransform(ODE, IC, y(x))

Parameters

 ODE - a linear ordinary differential equation IC - set; a set of two initial conditions y - name; the dependent variable x - name; the independent variable

Description

 • The ByLaplaceTransform(ODE, IC, y(x)) command finds the solution of a linear ordinary differential equation ODE with initial conditions IC by using the Laplace transform.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right):$
 > $\mathrm{ode1}≔\mathrm{diff}\left(x\left(t\right),t,t\right)+2\mathrm{diff}\left(x\left(t\right),t\right)+2x\left(t\right)=0$
 ${\mathrm{ode1}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{2}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{2}{}{x}{}\left({t}\right){=}{0}$ (1)
 > $\mathrm{ic1}≔\left\{\mathrm{eval}\left(\mathrm{diff}\left(x\left(t\right),t\right),t=0\right)=-2,x\left(0\right)=1\right\}$
 ${\mathrm{ic1}}{≔}\left\{\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}{\phantom{\left\{{t}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}}{\left\{{t}{=}{0}\right\}}{=}{-2}{,}{x}{}\left({0}\right){=}{1}\right\}$ (2)
 > $\mathrm{ByLaplaceTransform}\left(\mathrm{ode1},\mathrm{ic1},x\left(t\right)\right)$
 ${x}{}\left({t}\right){=}{{ⅇ}}^{{-}{t}}{}{\mathrm{cos}}{}\left({t}\right){-}{{ⅇ}}^{{-}{t}}{}{\mathrm{sin}}{}\left({t}\right)$ (3)
 > $\mathrm{ode2}≔\mathrm{diff}\left(x\left(t\right),t,t\right)+\mathrm{diff}\left(x\left(t\right),t\right)-6x\left(t\right)=\mathrm{sin}\left(t\right)$
 ${\mathrm{ode2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){-}{6}{}{x}{}\left({t}\right){=}{\mathrm{sin}}{}\left({t}\right)$ (4)
 > $\mathrm{ic2}≔\left\{\mathrm{eval}\left(\mathrm{diff}\left(x\left(t\right),t\right),t=0\right)=2,x\left(0\right)=-3\right\}$
 ${\mathrm{ic2}}{≔}\left\{\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}{\phantom{\left\{{t}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}}{\left\{{t}{=}{0}\right\}}{=}{2}{,}{x}{}\left({0}\right){=}{-3}\right\}$ (5)
 > $\mathrm{ByLaplaceTransform}\left(\mathrm{ode1},\mathrm{ic2},x\left(t\right)\right)$
 ${x}{}\left({t}\right){=}{-}{3}{}{{ⅇ}}^{{-}{t}}{}{\mathrm{cos}}{}\left({t}\right){-}{{ⅇ}}^{{-}{t}}{}{\mathrm{sin}}{}\left({t}\right)$ (6)
 > $\mathrm{ode3}≔\mathrm{diff}\left(x\left(t\right),t,t\right)+4\mathrm{diff}\left(x\left(t\right),t\right)+4x\left(t\right)=\mathrm{exp}\left(2t\right)$
 ${\mathrm{ode3}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{4}{}{x}{}\left({t}\right){=}{{ⅇ}}^{{2}{}{t}}$ (7)
 > $\mathrm{ic3}≔\left\{\mathrm{eval}\left(\mathrm{diff}\left(x\left(t\right),t\right),t=0\right)=-2,x\left(0\right)=1\right\}$
 ${\mathrm{ic3}}{≔}\left\{\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}{\phantom{\left\{{t}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}}{\left\{{t}{=}{0}\right\}}{=}{-2}{,}{x}{}\left({0}\right){=}{1}\right\}$ (8)
 > $\mathrm{ByLaplaceTransform}\left(\mathrm{ode1},\mathrm{ic3},x\left(t\right)\right)$
 ${x}{}\left({t}\right){=}{{ⅇ}}^{{-}{t}}{}{\mathrm{cos}}{}\left({t}\right){-}{{ⅇ}}^{{-}{t}}{}{\mathrm{sin}}{}\left({t}\right)$ (9)
 > $\mathrm{ode4}≔\mathrm{diff}\left(x\left(t\right),t,t\right)-6\mathrm{diff}\left(x\left(t\right),t\right)+13x\left(t\right)=t$
 ${\mathrm{ode4}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){-}{6}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{13}{}{x}{}\left({t}\right){=}{t}$ (10)
 > $\mathrm{ic4}≔\left\{\mathrm{eval}\left(\mathrm{diff}\left(x\left(t\right),t\right),t=0\right)=-2,x\left(0\right)=1\right\}$
 ${\mathrm{ic4}}{≔}\left\{\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}{\phantom{\left\{{t}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}}{\left\{{t}{=}{0}\right\}}{=}{-2}{,}{x}{}\left({0}\right){=}{1}\right\}$ (11)
 > $\mathrm{ByLaplaceTransform}\left(\mathrm{ode1},\mathrm{ic4},x\left(t\right)\right)$
 ${x}{}\left({t}\right){=}{{ⅇ}}^{{-}{t}}{}{\mathrm{cos}}{}\left({t}\right){-}{{ⅇ}}^{{-}{t}}{}{\mathrm{sin}}{}\left({t}\right)$ (12)
 > $\mathrm{ode5}≔\mathrm{diff}\left(x\left(t\right),t,t\right)+4\mathrm{diff}\left(x\left(t\right),t\right)+4x\left(t\right)=\mathrm{exp}\left(-2t\right)$
 ${\mathrm{ode5}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{4}{}{x}{}\left({t}\right){=}{{ⅇ}}^{{-}{2}{}{t}}$ (13)
 > $\mathrm{ic5}≔\left\{\mathrm{eval}\left(\mathrm{diff}\left(x\left(t\right),t\right),t=0\right)=-2,x\left(0\right)=2\right\}$
 ${\mathrm{ic5}}{≔}\left\{\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}{\phantom{\left\{{t}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}}{\left\{{t}{=}{0}\right\}}{=}{-2}{,}{x}{}\left({0}\right){=}{2}\right\}$ (14)
 > $\mathrm{ByLaplaceTransform}\left(\mathrm{ode1},\mathrm{ic5},x\left(t\right)\right)$
 ${x}{}\left({t}\right){=}{2}{}{{ⅇ}}^{{-}{t}}{}{\mathrm{cos}}{}\left({t}\right)$ (15)

Compatibility

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