Solving Linear ODEs - Maple Programming Help

Home : Support : Online Help : Mathematics : Differential Equations : Classifying ODEs : High Order : odeadvisor/linear_ODEs

Solving Linear ODEs

Description

 • The general form of the linear ODE is given by:

$\mathrm{linear_ODE}≔{A}_{0}y\left(x\right)+{A}_{1}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+{A}_{2}\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)+\mathrm{...}=F\left(x\right)$

 where the coefficients ${A}_{n}$ can be functions of $x$, see Differentialgleichungen, by E. Kamke, p. 69. Roughly speaking, there is no general method for solving the most general linear ODE of differential order greater than one. However, this is an active research area and there are many solving schemes which are applicable when the linear ODE satisfies certain conditions. In all the cases, if the method is applicable and the ODE is of second order, the ODE can be integrated to the end; otherwise, its order can be reduced by one or more, depending on the case. A summary of the methods implemented in dsolve for linear ODEs is as follows:
 – the ODE is exact (see odeadvisor, exact_linear);
 – the coefficients ${A}_{n}$ are rational functions and the ODE has exponential solutions (see DEtools, expsols);
 – the ODE has liouvillian solutions (see DEtools, kovacicsols);
 – the ODE has three regular singular points (see DEtools, RiemannPsols).
 – the ODE has simple symmetries of the form $\left[0,F\left(x\right)\right]$ (see odeadvisor, sym_Fx);
 – the ODE has special functions" solutions (see odeadvisor, classifications for second order ODEs).

Examples

The most general exact linear non-homogeneous ODE of second order; this case is solvable (see odeadvisor, exact_linear):

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (1)
 > $\mathrm{ode1}≔\frac{ⅆ}{ⅆx}\left(\frac{ⅆ}{ⅆx}y\left(x\right)=A\left(x\right)y\left(x\right)+B\left(x\right)\right)$
 ${\mathrm{ode1}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{A}{}\left({x}\right)\right){}{y}{}\left({x}\right){+}{A}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}\frac{{ⅆ}}{{ⅆ}{x}}{}{B}{}\left({x}\right)$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{ode1},y\left(x\right)\right)$
 $\left[\left[{\mathrm{_2nd_order}}{,}{\mathrm{_exact}}{,}{\mathrm{_linear}}{,}{\mathrm{_nonhomogeneous}}\right]\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{ode1},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}\left({\mathrm{_C2}}{+}{∫}\left({\mathrm{_C1}}{+}{B}{}\left({x}\right)\right){}{{ⅇ}}^{{∫}\left({-}{A}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){}{{ⅇ}}^{{-}\left({∫}\left({-}{A}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}$ (4)

Exponential solutions for a third order linear ODE .

 > $\mathrm{ode2}≔\left({x}^{2}+x\right)\left(\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)\right)-\left({x}^{2}-2\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-\left(x+2\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)=0$
 ${\mathrm{ode2}}{≔}\left({{x}}^{{2}}{+}{x}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{y}{}\left({x}\right)\right){-}\left({{x}}^{{2}}{-}{2}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){-}\left({x}{+}{2}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){=}{0}$ (5)
 > $\mathrm{dsolve}\left(\mathrm{ode2}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{+}{\mathrm{_C2}}{}{\mathrm{ln}}{}\left({x}\right){+}{\mathrm{_C3}}{}{{ⅇ}}^{{x}}$ (6)

An example of an ODE with regular singular points

 > $\mathrm{ode3}≔x\left(1-x\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)+\left(c-\left(a+b+1\right)x\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)-aby\left(x\right)$
 ${\mathrm{ode3}}{≔}{x}{}\left({1}{-}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){+}\left({c}{-}\left({a}{+}{b}{+}{1}\right){}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){-}{a}{}{b}{}{y}{}\left({x}\right)$ (7)
 > $\mathrm{dsolve}\left(\mathrm{ode3}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{hypergeom}}{}\left(\left[{a}{,}{b}\right]{,}\left[{c}\right]{,}{x}\right){+}{\mathrm{_C2}}{}{{x}}^{{-}{c}{+}{1}}{}{\mathrm{hypergeom}}{}\left(\left[{b}{-}{c}{+}{1}{,}{a}{-}{c}{+}{1}\right]{,}\left[{2}{-}{c}\right]{,}{x}\right)$ (8)

An example for which symmetries of the form $\left[0,F\left(x\right)\right]$ can be found (see odeadvisor, sym_Fx)

 > $\mathrm{ode4}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)=\mathrm{ln}\left(x\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+y\left(x\right)\left(1+\mathrm{ln}\left(x\right)\right)$
 ${\mathrm{ode4}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}{\mathrm{ln}}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{y}{}\left({x}\right){}\left({1}{+}{\mathrm{ln}}{}\left({x}\right)\right)$ (9)
 > $\mathrm{odeadvisor}\left(\mathrm{ode4}\right)$
 $\left[\left[{\mathrm{_2nd_order}}{,}{\mathrm{_with_linear_symmetries}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_linear}}{,}{\mathrm{_with_symmetry_\left[0,F\left(x\right)\right]}}\right]\right]$ (10)
 > $\mathrm{dsolve}\left(\mathrm{ode4}\right)$
 ${y}{}\left({x}\right){=}\left(\left({∫}\frac{{{x}}^{{x}}}{{{ⅇ}}^{{x}}{}{\left({{ⅇ}}^{{-}{x}}\right)}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){}{\mathrm{_C1}}{+}{\mathrm{_C2}}\right){}{{ⅇ}}^{{-}{x}}$ (11)

Some ODEs with special function solutions (see odeadvisor, second order ODEs).

Bessel ODE.

 > $\mathrm{ode5}≔{x}^{2}\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)+x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+\left({x}^{2}-{n}^{2}\right)y\left(x\right)=0$
 ${\mathrm{ode5}}{≔}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){}{{x}}^{{2}}{+}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){}{x}{+}\left({-}{{n}}^{{2}}{+}{{x}}^{{2}}\right){}{y}{}\left({x}\right){=}{0}$ (12)
 > $\mathrm{odeadvisor}\left(\mathrm{ode5}\right)$
 $\left[{\mathrm{_Bessel}}\right]$ (13)
 > $\mathrm{dsolve}\left(\mathrm{ode5}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{BesselJ}}{}\left({n}{,}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{BesselY}}{}\left({n}{,}{x}\right)$ (14)

Complete Elliptic Integral ODE.

 > $\mathrm{ode6}≔\frac{ⅆ}{ⅆx}\left(x\left(1-{x}^{2}\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)\right)-xy\left(x\right)=0$
 ${\mathrm{ode6}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){-}{2}{}{{x}}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{x}{}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){-}{x}{}{y}{}\left({x}\right){=}{0}$ (15)
 > $\mathrm{odeadvisor}\left(\mathrm{ode6}\right)$
 $\left[\left[{\mathrm{_elliptic}}{,}{\mathrm{_class_I}}\right]\right]$ (16)
 > $\mathrm{dsolve}\left(\mathrm{ode6}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{EllipticK}}{}\left({x}\right){+}{\mathrm{_C2}}{}{\mathrm{EllipticCK}}{}\left({x}\right)$ (17)

Gegenbauer ODE.

 > $\mathrm{ode7}≔\left({x}^{2}-1\right)\left(\frac{ⅆ}{ⅆx}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)\right)-\left(2m+3\right)x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+\mathrm{λ}y\left(x\right)=0$
 ${\mathrm{ode7}}{≔}\left({{x}}^{{2}}{-}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){-}\left({2}{}{m}{+}{3}\right){}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{\mathrm{λ}}{}{y}{}\left({x}\right){=}{0}$ (18)
 > $\mathrm{odeadvisor}\left(\mathrm{ode7}\right)$
 $\left[{\mathrm{_Gegenbauer}}\right]$ (19)
 > $\mathrm{dsolve}\left(\mathrm{ode7}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\left({{x}}^{{2}}{-}{1}\right)}^{\frac{{5}}{{4}}{+}\frac{{1}}{{2}}{}{m}}{}{\mathrm{LegendreP}}{}\left(\sqrt{{{m}}^{{2}}{-}{\mathrm{λ}}{+}{4}{}{m}{+}{4}}{-}\frac{{1}}{{2}}{,}\frac{{5}}{{2}}{+}{m}{,}{x}\right){+}{\mathrm{_C2}}{}{\left({{x}}^{{2}}{-}{1}\right)}^{\frac{{5}}{{4}}{+}\frac{{1}}{{2}}{}{m}}{}{\mathrm{LegendreQ}}{}\left(\sqrt{{{m}}^{{2}}{-}{\mathrm{λ}}{+}{4}{}{m}{+}{4}}{-}\frac{{1}}{{2}}{,}\frac{{5}}{{2}}{+}{m}{,}{x}\right)$ (20)