 series - Maple Help

dsolve

find series solutions to ODE problems Calling Sequence dsolve(ODE, y(x), 'series') dsolve(ODE, y(x), 'series', x=pt) dsolve({ODE, ICs}, y(x), 'series') dsolve({sysODE, ICs}, {funcs}, 'series') dsolve(ODE, y(x), 'type=series') dsolve(ODE, y(x), 'type=series', x=pt) dsolve({ODE, ICs}, y(x), 'type=series') dsolve({sysODE, ICs}, {funcs}, 'type=series') Parameters

 ODE - ordinary differential equation y(x) - dependent variable (indeterminate function) ICs - initial conditions for y(x) and/or its derivatives sysODE - system of ODEs {funcs} - set with indeterminate functions pt - expansion point for series 'type=series' - to request a series solution Options

 • combined = true or false

If true, returns a single series.  The default is false. Description

 • The dsolve command uses several methods when trying to find a series solution to an ODE or a system of ODEs. When initial conditions or an expansion point are given, the series is calculated at the given point; otherwise, the series is calculated at the origin.
 • The first method used is a Newton iteration based on a paper of Keith Geddes. See the References section in this help page.
 • The second method involves a direct substitution to generate a system of equations, which may be solvable (by solve) to give a series.
 • The third method is the method of Frobenius for nth order linear DEs. See the References section in this help page.
 • If the aforementioned methods fail, the function invokes LinearFunctionalSystems[SeriesSolution].
 • The series solutions may be expressed as the sum of several series (based on the solution method), rather than as a single series. If a single series is desired, the optional argument combined should be provided to assure that the output is provided in a single series data structure. Examples

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

When the initial conditions are not given, the answer is expressed in terms of the indeterminate function and its derivatives evaluated at the origin.

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\left\{\mathrm{ode}\right\},y\left(t\right),\mathrm{type}='\mathrm{series}'\right)$
 ${\mathrm{ans}}{≔}{y}{}\left({t}\right){=}{y}{}\left({0}\right){+}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){}{t}{-}\frac{{1}}{{2}}{}{{\mathrm{D}}{}\left({y}\right){}\left({0}\right)}^{{2}}{}{{t}}^{{2}}{+}\frac{{1}}{{3}}{}{{\mathrm{D}}{}\left({y}\right){}\left({0}\right)}^{{3}}{}{{t}}^{{3}}{-}\frac{{1}}{{4}}{}{{\mathrm{D}}{}\left({y}\right){}\left({0}\right)}^{{4}}{}{{t}}^{{4}}{+}\frac{{1}}{{5}}{}{{\mathrm{D}}{}\left({y}\right){}\left({0}\right)}^{{5}}{}{{t}}^{{5}}{+}{O}{}\left({{t}}^{{6}}\right)$ (2)

If initial conditions are given, the series is calculated at that the given point:

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\left\{\mathrm{ode},y\left(a\right)=\mathrm{Y_a},\mathrm{D}\left(y\right)\left(a\right)=\mathrm{DY_a}\right\},y\left(t\right),\mathrm{type}='\mathrm{series}'\right)$
 ${\mathrm{ans}}{≔}{y}{}\left({t}\right){=}{\mathrm{Y_a}}{+}{\mathrm{DY_a}}{}\left({t}{-}{a}\right){-}\frac{{1}}{{2}}{}{{\mathrm{DY_a}}}^{{2}}{}{\left({t}{-}{a}\right)}^{{2}}{+}\frac{{1}}{{3}}{}{{\mathrm{DY_a}}}^{{3}}{}{\left({t}{-}{a}\right)}^{{3}}{-}\frac{{1}}{{4}}{}{{\mathrm{DY_a}}}^{{4}}{}{\left({t}{-}{a}\right)}^{{4}}{+}\frac{{1}}{{5}}{}{{\mathrm{DY_a}}}^{{5}}{}{\left({t}{-}{a}\right)}^{{5}}{+}{O}{}\left({\left({t}{-}{a}\right)}^{{6}}\right)$ (3)

Alternatively, an expansion point can be provided, which is most useful when initial conditions cannot be given:

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\left(1-{t}^{2}\right)\left(\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}y\left(t\right)\right)-2ty\left(t\right)-y\left(t\right),y\left(t\right),'\mathrm{series}',t=1\right)$
 ${\mathrm{ans}}{≔}{y}{}\left({t}\right){=}\mathrm{c__1}{}\left({t}{-}{1}\right){}\left({1}{-}\frac{{3}}{{4}}{}\left({t}{-}{1}\right){+}\frac{{7}}{{48}}{}{\left({t}{-}{1}\right)}^{{2}}{+}\frac{{1}}{{128}}{}{\left({t}{-}{1}\right)}^{{3}}{-}\frac{{157}}{{15360}}{}{\left({t}{-}{1}\right)}^{{4}}{+}\frac{{3371}}{{921600}}{}{\left({t}{-}{1}\right)}^{{5}}{+}{O}{}\left({\left({t}{-}{1}\right)}^{{6}}\right)\right){+}\mathrm{c__2}{}\left({\mathrm{ln}}{}\left({t}{-}{1}\right){}\left({-}\frac{{3}}{{2}}{}\left({t}{-}{1}\right){+}\frac{{9}}{{8}}{}{\left({t}{-}{1}\right)}^{{2}}{-}\frac{{7}}{{32}}{}{\left({t}{-}{1}\right)}^{{3}}{-}\frac{{3}}{{256}}{}{\left({t}{-}{1}\right)}^{{4}}{+}\frac{{157}}{{10240}}{}{\left({t}{-}{1}\right)}^{{5}}{+}{O}{}\left({\left({t}{-}{1}\right)}^{{6}}\right)\right){+}\left({1}{-}\frac{{29}}{{16}}{}{\left({t}{-}{1}\right)}^{{2}}{+}\frac{{21}}{{32}}{}{\left({t}{-}{1}\right)}^{{3}}{-}\frac{{131}}{{3072}}{}{\left({t}{-}{1}\right)}^{{4}}{-}\frac{{2219}}{{102400}}{}{\left({t}{-}{1}\right)}^{{5}}{+}{O}{}\left({\left({t}{-}{1}\right)}^{{6}}\right)\right)\right)$ (4)

The above solution is expressed as a sum of several series. The following calling sequence provides the combined form:

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\left(1-{t}^{2}\right)\left(\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}y\left(t\right)\right)-2ty\left(t\right)-y\left(t\right),y\left(t\right),'\mathrm{series}','\mathrm{combined}',t=1\right)$
 ${\mathrm{ans}}{≔}{y}{}\left({t}\right){=}\mathrm{c__2}{+}\left(\mathrm{c__1}{-}\frac{{3}{}\mathrm{c__2}{}{\mathrm{ln}}{}\left({t}{-}{1}\right)}{{2}}\right){}\left({t}{-}{1}\right){+}\left({-}\frac{{3}{}\mathrm{c__1}}{{4}}{+}\mathrm{c__2}{}\left(\frac{{9}{}{\mathrm{ln}}{}\left({t}{-}{1}\right)}{{8}}{-}\frac{{29}}{{16}}\right)\right){}{\left({t}{-}{1}\right)}^{{2}}{+}\left(\frac{{7}{}\mathrm{c__1}}{{48}}{+}\mathrm{c__2}{}\left({-}\frac{{7}{}{\mathrm{ln}}{}\left({t}{-}{1}\right)}{{32}}{+}\frac{{21}}{{32}}\right)\right){}{\left({t}{-}{1}\right)}^{{3}}{+}\left(\frac{\mathrm{c__1}}{{128}}{+}\mathrm{c__2}{}\left({-}\frac{{3}{}{\mathrm{ln}}{}\left({t}{-}{1}\right)}{{256}}{-}\frac{{131}}{{3072}}\right)\right){}{\left({t}{-}{1}\right)}^{{4}}{+}\left({-}\frac{{157}{}\mathrm{c__1}}{{15360}}{+}\mathrm{c__2}{}\left(\frac{{157}{}{\mathrm{ln}}{}\left({t}{-}{1}\right)}{{10240}}{-}\frac{{2219}}{{102400}}\right)\right){}{\left({t}{-}{1}\right)}^{{5}}{+}{O}{}\left({\left({t}{-}{1}\right)}^{{6}}\right)$ (5)

The order of the series expansion (default = 6) can be changed using (an environment variable - see Order). For example,

 > $\mathrm{Order}≔3$
 ${\mathrm{Order}}{≔}{3}$ (6)

An example with a system of ODEs.

 > $\mathrm{sys}≔\left\{\frac{ⅆ}{ⅆt}y\left(t\right)=-x\left(t\right),\frac{ⅆ}{ⅆt}x\left(t\right)=y\left(t\right)\right\}$
 ${\mathrm{sys}}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){=}{y}{}\left({t}\right){,}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}{-}{x}{}\left({t}\right)\right\}$ (7)
 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{sys}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∪\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left\{x\left(0\right)=A,y\left(0\right)=B\right\},\left\{x\left(t\right),y\left(t\right)\right\},\mathrm{type}='\mathrm{series}'\right)$
 ${\mathrm{ans}}{≔}\left\{{x}{}\left({t}\right){=}{A}{+}{B}{}{t}{-}\frac{{1}}{{2}}{}{A}{}{{t}}^{{2}}{+}{O}{}\left({{t}}^{{3}}\right){,}{y}{}\left({t}\right){=}{B}{-}{A}{}{t}{-}\frac{{1}}{{2}}{}{B}{}{{t}}^{{2}}{+}{O}{}\left({{t}}^{{3}}\right)\right\}$ (8)

An example solved by LinearFunctionalSystems[SeriesSolution].

 > $\mathrm{sys}≔\left[\frac{ⅆ}{ⅆx}\mathrm{y1}\left(x\right)-\mathrm{y1}\left(x\right)+x\mathrm{y2}\left(x\right)={x}^{3},x\left(\frac{ⅆ}{ⅆx}\mathrm{y2}\left(x\right)\right)-2\mathrm{y2}\left(x\right)\right]$
 ${\mathrm{sys}}{≔}\left[\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y1}}{}\left({x}\right){-}{\mathrm{y1}}{}\left({x}\right){+}{x}{}{\mathrm{y2}}{}\left({x}\right){=}{{x}}^{{3}}{,}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{y2}}{}\left({x}\right)\right){-}{2}{}{\mathrm{y2}}{}\left({x}\right)\right]$ (9)
 > $\mathrm{vars}≔\left[\mathrm{y1}\left(x\right),\mathrm{y2}\left(x\right)\right]$
 ${\mathrm{vars}}{≔}\left[{\mathrm{y1}}{}\left({x}\right){,}{\mathrm{y2}}{}\left({x}\right)\right]$ (10)
 > $\mathrm{dsolve}\left(\left\{\mathrm{op}\left(\mathrm{sys}\right)\right\}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∪\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left\{\mathrm{y1}\left(0\right)=13\right\},\mathrm{vars},'\mathrm{series}'\right)$
 $\left\{{\mathrm{y1}}{}\left({x}\right){=}{13}{+}{13}{}{x}{+}\frac{{13}}{{2}}{}{{x}}^{{2}}{+}{O}{}\left({{x}}^{{3}}\right){,}{\mathrm{y2}}{}\left({x}\right){=}\frac{{{\mathrm{D}}}^{\left({2}\right)}{}\left({\mathrm{y2}}\right){}\left({0}\right)}{{2}}{}{{x}}^{{2}}{+}{O}{}\left({{x}}^{{3}}\right)\right\}$ (11) References

 Forsyth, A.R. Theory of Differential Equations. Cambridge: University Press, 1906. pp. 78-90
 Geddes, Keith.  "Convergence Behaviour of the Newton Iteration for First Order Differential Equations". Proceedings of EUROSAM '79. pp.189-199.
 Ince, E.L. Ordinary Differential Equations. Dover Publications, 1956. pp. 398-406. Compatibility

 • The dsolve command was updated in Maple 2023.
 • The combined option was introduced in Maple 2023.