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 2nd order 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 2nd order linear ordinary differential equation ODE with initial conditions IC by using the Laplace transform.
 • The coefficients of y(x) and its derivatives must be constant.
 • Use the option output=steps to make this command return an annotated step-by-step solution.  Further control over the format and display of the step-by-step solution is available using the options described in Student:-Basics:-OutputStepsRecord.  The options supported by that command can be passed to this one.

Examples

 > $\mathrm{with}\left({{\mathrm{Student}}_{\mathrm{ODEs}}}_{\mathrm{Solve}}\right):$
 > $\mathrm{ode1}≔\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}x\left(t\right)+2\left(\frac{ⅆ}{ⅆt}x\left(t\right)\right)+2x\left(t\right)=0$
 ${\mathrm{ode1}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{2}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{2}{}{x}{}\left({t}\right){=}{0}$ (1)
 > $\mathrm{ic1}≔\left\{\genfrac{}{}{0}{}{\frac{ⅆ}{ⅆt}x\left(t\right)}{\phantom{t=0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{ⅆ}{ⅆt}x\left(t\right)}}{t=0}=-2,x\left(0\right)=1\right\}$
 ${\mathrm{ic1}}{≔}\left\{\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}{\phantom{\left\{{t}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}}{\left\{{t}{=}{0}\right\}}{=}{-2}{,}{x}{}\left({0}\right){=}{1}\right\}$ (2)
 > $\mathrm{sol1}≔\mathrm{ByLaplaceTransform}\left(\mathrm{ode1},\mathrm{ic1},x\left(t\right)\right)$
 ${\mathrm{sol1}}{≔}{x}{}\left({t}\right){=}{{ⅇ}}^{{-}{t}}{}{\mathrm{cos}}{}\left({t}\right){-}{{ⅇ}}^{{-}{t}}{}{\mathrm{sin}}{}\left({t}\right)$ (3)
 > $\mathrm{ode2}≔\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}x\left(t\right)+\frac{ⅆ}{ⅆt}x\left(t\right)-6x\left(t\right)=\mathrm{sin}\left(t\right)$
 ${\mathrm{ode2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){-}{6}{}{x}{}\left({t}\right){=}{\mathrm{sin}}{}\left({t}\right)$ (4)
 > $\mathrm{ic2}≔\left\{\genfrac{}{}{0}{}{\frac{ⅆ}{ⅆt}x\left(t\right)}{\phantom{t=0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{ⅆ}{ⅆt}x\left(t\right)}}{t=0}=2,x\left(0\right)=-3\right\}$
 ${\mathrm{ic2}}{≔}\left\{\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}{\phantom{\left\{{t}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}}{\left\{{t}{=}{0}\right\}}{=}{2}{,}{x}{}\left({0}\right){=}{-3}\right\}$ (5)
 > $\mathrm{sol2}≔\mathrm{ByLaplaceTransform}\left(\mathrm{ode2},\mathrm{ic2},x\left(t\right)\right)$
 ${\mathrm{sol2}}{≔}{x}{}\left({t}\right){=}{-}\frac{{{ⅇ}}^{{-}{3}{}{t}}{}\left(\left({\mathrm{cos}}{}\left({t}\right){+}{7}{}{\mathrm{sin}}{}\left({t}\right)\right){}{{ⅇ}}^{{3}{}{t}}{+}{68}{}{{ⅇ}}^{{5}{}{t}}{+}{81}\right)}{{50}}$ (6)
 > $\mathrm{ode3}≔\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}x\left(t\right)+4\left(\frac{ⅆ}{ⅆt}x\left(t\right)\right)+4x\left(t\right)={ⅇ}^{2t}$
 ${\mathrm{ode3}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{4}{}{x}{}\left({t}\right){=}{{ⅇ}}^{{2}{}{t}}$ (7)
 > $\mathrm{ic3}≔\left\{\genfrac{}{}{0}{}{\frac{ⅆ}{ⅆt}x\left(t\right)}{\phantom{t=0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{ⅆ}{ⅆt}x\left(t\right)}}{t=0}=-2,x\left(0\right)=1\right\}$
 ${\mathrm{ic3}}{≔}\left\{\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}{\phantom{\left\{{t}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}}{\left\{{t}{=}{0}\right\}}{=}{-2}{,}{x}{}\left({0}\right){=}{1}\right\}$ (8)
 > $\mathrm{sol3}≔\mathrm{ByLaplaceTransform}\left(\mathrm{ode3},\mathrm{ic3},x\left(t\right)\right)$
 ${\mathrm{sol3}}{≔}{x}{}\left({t}\right){=}\frac{\left({-}{4}{}{t}{+}{15}\right){}{{ⅇ}}^{{-}{2}{}{t}}}{{16}}{+}\frac{{{ⅇ}}^{{2}{}{t}}}{{16}}$ (9)
 > $\mathrm{ode4}≔\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}x\left(t\right)-6\left(\frac{ⅆ}{ⅆt}x\left(t\right)\right)+13x\left(t\right)=t$
 ${\mathrm{ode4}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){-}{6}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{13}{}{x}{}\left({t}\right){=}{t}$ (10)
 > $\mathrm{ic4}≔\left\{\genfrac{}{}{0}{}{\frac{ⅆ}{ⅆt}x\left(t\right)}{\phantom{t=0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{ⅆ}{ⅆt}x\left(t\right)}}{t=0}=-2,x\left(0\right)=1\right\}$
 ${\mathrm{ic4}}{≔}\left\{\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}{\phantom{\left\{{t}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}}{\left\{{t}{=}{0}\right\}}{=}{-2}{,}{x}{}\left({0}\right){=}{1}\right\}$ (11)
 > $\mathrm{sol4}≔\mathrm{ByLaplaceTransform}\left(\mathrm{ode4},\mathrm{ic4},x\left(t\right)\right)$
 ${\mathrm{sol4}}{≔}{x}{}\left({t}\right){=}\frac{\left({163}{}{\mathrm{cos}}{}\left({2}{}{t}\right){-}{420}{}{\mathrm{sin}}{}\left({2}{}{t}\right)\right){}{{ⅇ}}^{{3}{}{t}}}{{169}}{+}\frac{{t}}{{13}}{+}\frac{{6}}{{169}}$ (12)
 > $\mathrm{ode5}≔\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}x\left(t\right)+4\left(\frac{ⅆ}{ⅆt}x\left(t\right)\right)+4x\left(t\right)={ⅇ}^{-2t}$
 ${\mathrm{ode5}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{4}{}{x}{}\left({t}\right){=}{{ⅇ}}^{{-}{2}{}{t}}$ (13)
 > $\mathrm{ic5}≔\left\{\genfrac{}{}{0}{}{\frac{ⅆ}{ⅆt}x\left(t\right)}{\phantom{t=0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{ⅆ}{ⅆt}x\left(t\right)}}{t=0}=-2,x\left(0\right)=2\right\}$
 ${\mathrm{ic5}}{≔}\left\{\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}{\phantom{\left\{{t}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)}}{\left\{{t}{=}{0}\right\}}{=}{-2}{,}{x}{}\left({0}\right){=}{2}\right\}$ (14)
 > $\mathrm{sol5}≔\mathrm{ByLaplaceTransform}\left(\mathrm{ode5},\mathrm{ic5},x\left(t\right)\right)$
 ${\mathrm{sol5}}{≔}{x}{}\left({t}\right){=}\frac{{{ⅇ}}^{{-}{2}{}{t}}{}{\left({t}{+}{2}\right)}^{{2}}}{{2}}$ (15)

Compatibility

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