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powseries

 powsolve
 solve linear differential equations as power series

 Calling Sequence powsolve(sys)

Parameters

 sys - set or expression sequence containing a linear differential equation and optional initial conditions

Description

 • The function powsolve solves a linear differential equation for which initial conditions do not have to be specified.
 • All the initial conditions must be at zero.
 • Derivatives are denoted by applying $\mathrm{D}$ to the function name. For example, the second derivative of $y$ at 0 is $\mathrm{D}\left(\mathrm{D}\left(y\right)\right)\left(0\right)$.
 • The solution returned is a formal power series that represents the infinite series solution.
 • In some cases, after assigning the name a to the output from the powsolve command, you can enter the command a(_k) to output a recurrence relation for the power series solution.  See examples below.
 • The command with(powseries,powsolve) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{powseries}\right):$
 > $a≔\mathrm{powsolve}\left(\frac{ⅆ}{ⅆx}y\left(x\right)=y\left(x\right),y\left(0\right)=1\right):$
 > $\mathrm{tpsform}\left(a,x\right)$
 ${1}{+}{x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}{+}\frac{{1}}{{24}}{}{{x}}^{{4}}{+}\frac{{1}}{{120}}{}{{x}}^{{5}}{+}{\mathrm{O}}\left({{x}}^{{6}}\right)$ (1)
 > $a\left(\mathrm{_k}\right)$
 $\frac{{a}{}\left({\mathrm{_k}}{-}{1}\right)}{{\mathrm{_k}}}$ (2)

second system

 > $v≔\mathrm{powsolve}\left(\left\{\frac{{ⅆ}^{4}}{ⅆ{x}^{4}}y\left(x\right)=y\left(x\right),y\left(0\right)=\frac{3}{2},\mathrm{D}\left(y\right)\left(0\right)=-\frac{1}{2},\mathrm{D}\left(\mathrm{D}\left(y\right)\right)\left(0\right)=-\frac{3}{2},\mathrm{D}\left(\mathrm{D}\left(\mathrm{D}\left(y\right)\right)\right)\left(0\right)=\frac{1}{2}\right\}\right):$
 > $\mathrm{tpsform}\left(v,x\right)$
 $\frac{{3}}{{2}}{-}\frac{{1}}{{2}}{}{x}{-}\frac{{3}}{{4}}{}{{x}}^{{2}}{+}\frac{{1}}{{12}}{}{{x}}^{{3}}{+}\frac{{1}}{{16}}{}{{x}}^{{4}}{-}\frac{{1}}{{240}}{}{{x}}^{{5}}{+}{\mathrm{O}}\left({{x}}^{{6}}\right)$ (3)