Solve a Partial Differential Equation Symbolically - Maple Help

Solve a Partial Differential Equation Symbolically

 Description Solve a partial differential equation (PDE) symbolically.

Enter a PDE.

 > $\left(\frac{{ⅆ}}{{ⅆ}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}\left({x}{,}{t}\right)\right){-}{k}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}\left({x}{,}{t}\right)\right){=}{0}$
 $\frac{{\partial }}{{\partial }{t}}{}{u}{}\left({x}{,}{t}\right){-}{k}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{u}{}\left({x}{,}{t}\right)\right){=}{0}$ (1)

Solve the PDE.

 > $\mathrm{pdsolve}\left(,{u}\left({x}{,}{t}\right)\right)$
 $\left({u}{}\left({x}{,}{t}\right){=}{\mathrm{_F1}}{}\left({x}\right){}{\mathrm{_F2}}{}\left({t}\right)\right){&where}\left[\left\{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{\mathrm{_F1}}{}\left({x}\right){=}{{\mathrm{_c}}}_{{1}}{}{\mathrm{_F1}}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{t}}{}{\mathrm{_F2}}{}\left({t}\right){=}{k}{}{{\mathrm{_c}}}_{{1}}{}{\mathrm{_F2}}{}\left({t}\right)\right\}\right]$ (2)

Build an explicit expression for the indeterminate function, if possible.

 > $\mathrm{PDEtools}\left[\mathrm{build}\right]\left(\right)$
 ${u}{}\left({x}{,}{t}\right){=}{{ⅇ}}^{\sqrt{{{\mathrm{_c}}}_{{1}}}{}{x}}{}{\mathrm{_C3}}{}{{ⅇ}}^{{k}{}{{\mathrm{_c}}}_{{1}}{}{t}}{}{\mathrm{_C1}}{+}\frac{{\mathrm{_C3}}{}{{ⅇ}}^{{k}{}{{\mathrm{_c}}}_{{1}}{}{t}}{}{\mathrm{_C2}}}{{{ⅇ}}^{\sqrt{{{\mathrm{_c}}}_{{1}}}{}{x}}}$ (3)
 Commands Used