DEtools/particularsol - Maple Help

DEtools

 particularsol
 find a particular solution to a nonlinear ODE, or a linear non-homogeneous ODE, without computing its general solution

 Calling Sequence particularsol(ODE, y(x)) particularsol(ODE_coeff_list, x)

Parameters

 ODE - nonlinear, or linear non-homogeneous ODE y(x) - dependent variable (required only when not obvious); can be any unknown function of one variable ODE_coeff_list - list of coefficients representing a non-homogeneous linear ODE (see DEtools[convertAlg]) x - independent variable

Description

 • The particularsol routine is used to find a particular solution for a nonlinear ordinary differential equation (ODE), or for a non-homogeneous linear ODE without computing the general solution to its homogeneous part.
 • For nonlinear ODEs, the approach used is to compute symmetries and from there compute invariant (particular) solutions.
 • For linear ODEs, particularsol tries, in sequence, to compute particular solutions of rational (see DEtools[ratsols]), exponential and d'Alembertian form (see LinearOperators[dAlembertianSolver]). If no particular solution is found, particularsol returns NULL.
 • When the input is an ODE, the output of particularsol is as dsolve's output, that is, an equation with the unknown $y\left(x\right)$ on the left-hand-side and the (particular) solution on the right-hand-side.
 • When the input is a list of the coefficients of $y\left(x\right)$ and its derivatives representing a linear ODE, for instance obtained from the ODE using DEtools[convertAlg], the output is not an equation but an expression representing the particular solution - see the examples.
 • In the case of a linear ODE, while particularsol is useful to compute a particular solution when the general solution of the homogeneous part of the ODE is not known, a particular solution can always be computed if that general solution is known - for that purpose use DEtools[varparam].
 • This function is part of the DEtools package, and so it can be used in the form particularsol(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[particularsol](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$

A third order nonlinear ODE

 > $\mathrm{ode}≔\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)-\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)y\left(x\right)+{\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)}^{2}=0$
 ${\mathrm{ode}}{≔}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{y}{}\left({x}\right){+}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{=}{0}$ (1)

No general solution is known for ode. A particular solution for it, however, depending on one arbitrary constant, is computable via

 > $\mathrm{particularsol}\left(\mathrm{ode}\right)$
 ${y}{}\left({x}\right){=}{0}{,}{y}{}\left({x}\right){=}\frac{{6}}{{-}{x}{+}\mathrm{c__1}}$ (2)

A second order non-homogeneous linear ODE whose homogeneous part is out of the scope of dsolve and a particular rational solution for it

 > $\mathrm{NH_ODE}≔133601600831056948930009536{x}^{3}+920184312063950707048658496{x}^{2}+\left(-4242610768571010035328624653+3711155578640470803611376{x}^{4}-369364944765014363775910745{x}^{3}-4028299868602335386591950537{x}^{2}-7123173270557941102013850561x\right)y\left(x\right)+\left(158668797683584879758300854-361937570512208809606272361{x}^{4}-743569265631618257116222299{x}^{3}+253413657274110944226342787{x}^{2}+698563894241740551690579371x\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+\left(-3195084416888717732807842-8089797769268184221773945x+14158747037070037877844107{x}^{2}+8089797769268184221773945{x}^{3}-10963662620181320145036265{x}^{4}\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)$
 ${\mathrm{NH_ODE}}{≔}{133601600831056948930009536}{}{{x}}^{{3}}{+}{920184312063950707048658496}{}{{x}}^{{2}}{+}\left({3711155578640470803611376}{}{{x}}^{{4}}{-}{369364944765014363775910745}{}{{x}}^{{3}}{-}{4028299868602335386591950537}{}{{x}}^{{2}}{-}{7123173270557941102013850561}{}{x}{-}{4242610768571010035328624653}\right){}{y}{}\left({x}\right){+}\left({-}{361937570512208809606272361}{}{{x}}^{{4}}{-}{743569265631618257116222299}{}{{x}}^{{3}}{+}{253413657274110944226342787}{}{{x}}^{{2}}{+}{698563894241740551690579371}{}{x}{+}{158668797683584879758300854}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{10963662620181320145036265}{}{{x}}^{{4}}{+}{8089797769268184221773945}{}{{x}}^{{3}}{+}{14158747037070037877844107}{}{{x}}^{{2}}{-}{8089797769268184221773945}{}{x}{-}{3195084416888717732807842}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (3)
 > $\mathrm{p_sol}≔\mathrm{particularsol}\left(\mathrm{NH_ODE}\right)$
 ${\mathrm{p_sol}}{≔}{y}{}\left({x}\right){=}{-}\frac{{4}{}{{x}}^{{4}}{}\left({9}{}{{x}}^{{4}}{+}{35}{}{{x}}^{{3}}{+}{49}{}{{x}}^{{2}}{+}{21}{}{x}{+}{6}\right)}{{\left({{x}}^{{2}}{-}{1}\right)}^{{2}}{}{\left({x}{-}{1}\right)}^{{5}}}$ (4)
 > $\mathrm{odetest}\left(\mathrm{p_sol},\mathrm{NH_ODE}\right)$
 ${0}$ (5)

Represent this non-homogeneous linear ODE using a list with the coefficients of y(x) and its derivatives

 > $\mathrm{coeff_list}≔{\mathrm{DEtools}}_{\mathrm{convertAlg}}\left(\mathrm{NH_ODE},y\left(x\right)\right)$
 ${\mathrm{coeff_list}}{≔}\left[\left[{3711155578640470803611376}{}{{x}}^{{4}}{-}{369364944765014363775910745}{}{{x}}^{{3}}{-}{4028299868602335386591950537}{}{{x}}^{{2}}{-}{7123173270557941102013850561}{}{x}{-}{4242610768571010035328624653}{,}{-}{361937570512208809606272361}{}{{x}}^{{4}}{-}{743569265631618257116222299}{}{{x}}^{{3}}{+}{253413657274110944226342787}{}{{x}}^{{2}}{+}{698563894241740551690579371}{}{x}{+}{158668797683584879758300854}{,}{-}{10963662620181320145036265}{}{{x}}^{{4}}{+}{8089797769268184221773945}{}{{x}}^{{3}}{+}{14158747037070037877844107}{}{{x}}^{{2}}{-}{8089797769268184221773945}{}{x}{-}{3195084416888717732807842}\right]{,}{-}{133601600831056948930009536}{}{{x}}^{{3}}{-}{920184312063950707048658496}{}{{x}}^{{2}}\right]$ (6)
 > $\mathrm{p_sol}≔\mathrm{particularsol}\left(\mathrm{coeff_list},x\right)$
 ${\mathrm{p_sol}}{≔}{-}\frac{{4}{}{{x}}^{{4}}{}\left({9}{}{{x}}^{{4}}{+}{35}{}{{x}}^{{3}}{+}{49}{}{{x}}^{{2}}{+}{21}{}{x}{+}{6}\right)}{{\left({{x}}^{{2}}{-}{1}\right)}^{{2}}{}{\left({x}{-}{1}\right)}^{{5}}}$ (7)

Note particularsol is not restricted to the computation of rational particular solutions

 > $\mathrm{ode}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)-y\left(x\right)x=\frac{\left(4{x}^{3}+1\right)\mathrm{ln}\left(x\right)}{x\sqrt{x}}$
 ${\mathrm{ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{y}{}\left({x}\right){}{x}{=}\frac{\left({4}{}{{x}}^{{3}}{+}{1}\right){}{\mathrm{ln}}{}\left({x}\right)}{{{x}}^{{3}}{{2}}}}$ (8)
 > $\mathrm{p_sol}≔\mathrm{particularsol}\left(\mathrm{ode}\right)$
 ${\mathrm{p_sol}}{≔}{y}{}\left({x}\right){=}{-}{4}{}\sqrt{{x}}{}{\mathrm{ln}}{}\left({x}\right)$ (9)
 > $\mathrm{odetest}\left(\mathrm{p_sol},\mathrm{ode}\right)$
 ${0}$ (10)