exact linear - Maple Help

Solving Exact Linear ODEs

Description

 • The general form of the exact, linear ODE is given by the following:
 > exact_linear_ode := diff(linear_ODE(x),x) = 0;
 ${\mathrm{exact_linear_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{linear_ODE}}{}\left({x}\right){=}{0}$ (1)
 where linearODE(x) is a linear ODE of any differential order; see Murphy, "Ordinary Differential Equations and their Solutions", p. 221. The order of these exact linear ODEs can be reduced since they are the total derivative of an ODE of one order lower. The reduced ODE is:
 > linear_ODE(x) + _C1;
 ${\mathrm{linear_ODE}}{}\left({x}\right){+}{\mathrm{_C1}}$ (2)

Examples

The most general exact linear non-homogeneous ODE of second order; this case is solvable.

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (3)
 > $\mathrm{ODE}≔\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right)=A\left(x\right)y\left(x\right)+B\left(x\right),x\right)$
 ${\mathrm{ODE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({x}\right)\right){}{y}{}\left({x}\right){+}{A}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{B}{}\left({x}\right)$ (4)
 > $\mathrm{odeadvisor}\left(\mathrm{ODE},y\left(x\right)\right)$
 $\left[\left[{\mathrm{_2nd_order}}{,}{\mathrm{_exact}}{,}{\mathrm{_linear}}{,}{\mathrm{_nonhomogeneous}}\right]\right]$ (5)
 > $\mathrm{dsolve}\left(\mathrm{ODE},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}\left({\mathrm{_C2}}{+}{\int }\left({\mathrm{_C1}}{+}{B}{}\left({x}\right)\right){}{{ⅇ}}^{{\int }{-}{A}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){}{{ⅇ}}^{{-}\left({\int }{-}{A}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}$ (6)

The general exact linear ODE of fourth order which can be reduced to an exact linear ODE of third order; this can be reduced to a second order ODE and the answer is expressed using DESol

 > $\mathrm{ODE}≔\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x,x\right)=A\left(x\right)y\left(x\right)+B\left(x\right)\mathrm{diff}\left(y\left(x\right),x\right)+F\left(x\right),x,x\right)$
 ${\mathrm{ODE}}{≔}\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({x}\right)\right){}{y}{}\left({x}\right){+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{A}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{B}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{B}{}\left({x}\right)\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{B}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{F}{}\left({x}\right)$ (7)
 > $\mathrm{odeadvisor}\left(\mathrm{ODE},y\left(x\right)\right)$
 $\left[\left[{\mathrm{_high_order}}{,}{\mathrm{_exact}}{,}{\mathrm{_linear}}{,}{\mathrm{_nonhomogeneous}}\right]\right]$ (8)
 > $\mathrm{dsolve}\left(\mathrm{ODE},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{DESol}}{}\left(\left\{{-}{A}{}\left({x}\right){}{\mathrm{_Y}}{}\left({x}\right){-}{B}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_Y}}{}\left({x}\right)\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_Y}}{}\left({x}\right){-}{\mathrm{_C2}}{-}{\mathrm{_C1}}{}{x}{-}{F}{}\left({x}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({x}\right)\right\}\right)$ (9)

The general exact linear ODE of fifth order which can be reduced to a first order linear ODE. This ODE can be solved to the end.

 > $\mathrm{ODE}≔\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right)=A\left(x\right)y\left(x\right)+B\left(x\right),x,x,x,x\right)$
 ${\mathrm{ODE}}{≔}\frac{{{ⅆ}}^{{5}}}{{ⅆ}{{x}}^{{5}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({x}\right)\right){}{y}{}\left({x}\right){+}{4}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{6}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({x}\right)\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{4}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{A}{}\left({x}\right)\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{A}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{B}{}\left({x}\right)$ (10)
 > $\mathrm{odeadvisor}\left(\mathrm{ODE},y\left(x\right)\right)$
 $\left[\left[{\mathrm{_high_order}}{,}{\mathrm{_fully}}{,}{\mathrm{_exact}}{,}{\mathrm{_linear}}\right]\right]$ (11)
 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{ODE},y\left(x\right)\right)$
 ${\mathrm{ans}}{≔}{y}{}\left({x}\right){=}\left({\mathrm{_C5}}{+}{\int }\left({4}{}{\mathrm{_C1}}{}{{x}}^{{3}}{+}{3}{}{\mathrm{_C2}}{}{{x}}^{{2}}{+}{2}{}{\mathrm{_C3}}{}{x}{+}{\mathrm{_C4}}{+}{B}{}\left({x}\right)\right){}{{ⅇ}}^{{\int }{-}{A}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){}{{ⅇ}}^{{-}\left({\int }{-}{A}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}$ (12)
 > $\mathrm{odetest}\left(\mathrm{ans},\mathrm{ODE}\right)$
 ${0}$ (13)