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DEtools

 DFactorsols
 find solutions of a linear ode using DFactor

 Calling Sequence DFactorsols(lode, v) DFactorsols(coeff_list, g, x)

Parameters

 lode - linear differential equation v - dependent variable of the lode coeff_list - list of coefficients of a linear ode g - right-hand side of equation x - independent variable of the lode

Description

 • The DFactorsols routine returns a basis of solutions of a linear ODE. Using bases of linear equations determined by DEtools[DFactor], it factorizes the equation into smaller parts.
 • There are two possible calling sequences. The first calling sequence accepts a linear differential equation in diff or $\mathrm{D}$ form as the first argument and the variable in the differential equation as the second argument.
 • The second form of the calling sequence accepts the list of coefficients of a linear ode as the first argument, the right-hand side of such an equation for the second argument, and the independent variable of the lode as the third argument. This form of calling sequence is convenient for programming with the DFactorsols routine.
 In the second calling sequence, the list of coefficients is given in order from low differential order to high differential order and does not include the nonhomogeneous term.
 • In the case of a homogeneous equation, a basis is returned in the form of a list. In the nonhomogeneous case, the return value is a two-element list, with the first element a basis for the homogeneous case and the second element a particular solution.
 • This function is part of the DEtools package, and so it can be used in the form DFactorsols(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[DFactorsols](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{ode}≔\left({t}^{2}+t\right)\left(\frac{{ⅆ}^{3}}{ⅆ{t}^{3}}z\left(t\right)\right)-\left({t}^{2}-2\right)\left(\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}z\left(t\right)\right)-\left(t+2\right)\left(\frac{ⅆ}{ⅆt}z\left(t\right)\right)=0:$
 > $\mathrm{DFactorsols}\left(\mathrm{ode},z\left(t\right)\right)$
 $\left[{1}{,}{\mathrm{ln}}{}\left({t}\right){,}{{ⅇ}}^{{t}}\right]$ (1)
 > $\mathrm{ode}≔\left({t}^{2}+t\right)\left(\frac{{ⅆ}^{3}}{ⅆ{t}^{3}}z\left(t\right)\right)-\left({t}^{2}-2\right)\left(\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}z\left(t\right)\right)-\left(t+2\right)\left(\frac{ⅆ}{ⅆt}z\left(t\right)\right)=3t{\left(t+1\right)}^{2}:$
 > $\mathrm{DFactorsols}\left(\mathrm{ode},z\left(t\right)\right)$
 $\left[\left[{1}{,}{\mathrm{ln}}{}\left({t}\right){,}{{ⅇ}}^{{t}}\right]{,}{-}\frac{{1}}{{3}}{}{{t}}^{{3}}{-}\frac{{3}}{{2}}{}{{t}}^{{2}}{-}{3}{}{t}\right]$ (2)
 > $\mathrm{ode}≔\left({t}^{2}+t\right){\mathrm{D}}^{\left(3\right)}\left(z\right)\left(t\right)-\left({t}^{2}-2\right){\mathrm{D}}^{\left(2\right)}\left(z\right)\left(t\right)-\left(t+2\right)\mathrm{D}\left(z\right)\left(t\right)=3t{\left(t+1\right)}^{2}:$
 > $\mathrm{DFactorsols}\left(\mathrm{ode},z\left(t\right)\right)$
 $\left[\left[{1}{,}{\mathrm{ln}}{}\left({t}\right){,}{{ⅇ}}^{{t}}\right]{,}{-}\frac{{1}}{{3}}{}{{t}}^{{3}}{-}\frac{{3}}{{2}}{}{{t}}^{{2}}{-}{3}{}{t}\right]$ (3)
 > $\mathrm{ode_list}≔\left[0,-\left(t+2\right),-\left({t}^{2}-2\right),{t}^{2}+t\right]:$
 > $\mathrm{DFactorsols}\left(\mathrm{ode_list},3t{\left(t+1\right)}^{2},t\right)$
 $\left[\left[{1}{,}{\mathrm{ln}}{}\left({t}\right){,}{{ⅇ}}^{{t}}\right]{,}{-}\frac{{1}}{{3}}{}{{t}}^{{3}}{-}\frac{{3}}{{2}}{}{{t}}^{{2}}{-}{3}{}{t}\right]$ (4)
 > $\mathrm{ode}≔\frac{{ⅆ}^{3}}{ⅆ{t}^{3}}z\left(t\right)-\left(\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}z\left(t\right)\right)-t\left(\frac{ⅆ}{ⅆt}z\left(t\right)\right)+tz\left(t\right):$
 > $\mathrm{DFactorsols}\left(\mathrm{ode},z\left(t\right)\right)$
 $\left[{{ⅇ}}^{{t}}{,}\left({\int }{\mathrm{AiryAi}}{}\left({t}\right){}{{ⅇ}}^{{-}{t}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}\right){}{{ⅇ}}^{{t}}{,}\left({\int }{\mathrm{AiryBi}}{}\left({t}\right){}{{ⅇ}}^{{-}{t}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}\right){}{{ⅇ}}^{{t}}\right]$ (5)
 > $\mathrm{ode}≔\frac{{ⅆ}^{3}}{ⅆ{t}^{3}}z\left(t\right)-\left(\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}z\left(t\right)\right)-t\left(\frac{ⅆ}{ⅆt}z\left(t\right)\right)+\left(t-1\right)z\left(t\right):$
 > $\mathrm{DFactorsols}\left(\mathrm{ode},z\left(t\right)\right)$
 $\left[{\mathrm{AiryAi}}{}\left({t}\right){,}{\mathrm{AiryBi}}{}\left({t}\right){,}{-}{\mathrm{AiryAi}}{}\left({t}\right){}\left({\int }{\mathrm{AiryBi}}{}\left({t}\right){}{{ⅇ}}^{{t}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}\right){+}\left({\int }{\mathrm{AiryAi}}{}\left({t}\right){}{{ⅇ}}^{{t}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}\right){}{\mathrm{AiryBi}}{}\left({t}\right)\right]$ (6)