Series Solutions

The following examples illustrate series solutions to various types of systems of differential equations. You can control the number of terms that appear in the solution by changing Series Order for ODE Solutions in the Maple Settings dialog. For most of the following examples, the series order has been set at 6.


$\blacktriangleright$ Solve ODE + Series

y = x
x = - y
x(0) = 0
y(0) = 1
(Specify t), Series solution is :
y$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = 1 - ${\frac{{1}}{{2}}}$t2 + ${\frac{{1}}{{24}}}$t4 + O$\left(\vphantom{ t^{6}}\right.$t6$\left.\vphantom{ t^{6}}\right)$
x$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = - t + ${\frac{{1}}{{6}}}$t3 - ${\frac{{1}}{{120}}}$t5 + O$\left(\vphantom{ t^{6}}\right.$t6$\left.\vphantom{ t^{6}}\right)$

 


y = y2 + 4
y(0) = - 2
(Specify t), Series solution is:
y$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = - 2 + 8t - 16t2 + ${\dfrac{{128}}{{3}}}$t3
             - ${\dfrac{{320}}{{3}}}$t4 + ${\dfrac{{4096}}{{15}}}$t5 + O$\left(\vphantom{ t^{6}}\right.$t6$\left.\vphantom{ t^{6}}\right)$

 


Dxxy1 - y1 = 0
y1(0) = 1
y1$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$ = 0
, Series solution is: y1$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = 1 + ${\dfrac{{1}}{{2}}}$x2 + ${\dfrac{{1}}{{%
24}}}$x4 + O$\left(\vphantom{ x^{6}}\right.$x6$\left.\vphantom{ x^{6}}\right)$

 


y = x
x = y
x(0) = 0
y(0) = 1
(Specify t), Series solution is:
x$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = t + ${\dfrac{{1}}{{6}}}$t3 + ${\dfrac{{1}}{{120}}}$t5 + O$\left(\vphantom{ t^{6}}\right.$t6$\left.\vphantom{ t^{6}}\right)$
y$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = 1 + ${\dfrac{{1}}{{2}}}$t2 + ${\dfrac{{1}}{{24}}}$t4 + O$\left(\vphantom{ t^{6}}\right.$t6$\left.\vphantom{ t^{6}}\right)$