90.32.1 problem 1
Internal
problem
ID
[25474]
Book
:
Ordinary
Differential
Equations.
By
William
Adkins
and
Mark
G
Davidson.
Springer.
NY.
2010.
ISBN
978-1-4614-3617-1
Section
:
Chapter
7.
Power
series
methods.
Exercises
at
page
551
Problem
number
:
1
Date
solved
:
Friday, October 03, 2025 at 12:01:55 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} t y^{\prime \prime }+2 \left (i t -k \right ) y^{\prime }-2 i k y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 174
Order:=6;
ode:=t*diff(diff(y(t),t),t)+2*(I*t-k)*diff(y(t),t)-2*I*k*y(t) = 0;
dsolve(ode,y(t),type='series',t=0);
\[
y = t^{2 k +1} c_1 \left (1-i t +\frac {-k -2}{2 k +3} t^{2}+\frac {i \left (k +3\right )}{6 k +9} t^{3}+\frac {\left (k +3\right ) \left (k +4\right )}{24 k^{2}+96 k +90} t^{4}-\frac {i \left (k +5\right ) \left (k +4\right )}{120 k^{2}+480 k +450} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1-i t +\frac {-k +1}{2 k -1} t^{2}+\frac {i \left (k -2\right )}{6 k -3} t^{3}+\frac {\left (k -2\right ) \left (k -3\right )}{24 k^{2}-48 k +18} t^{4}-\frac {i \left (k -4\right ) \left (k -3\right )}{120 k^{2}-240 k +90} t^{5}+\operatorname {O}\left (t^{6}\right )\right )
\]
✓ Mathematica. Time used: 0.002 (sec). Leaf size: 789
ode=t*D[y[t],{t,2}]+2*(I*t-k)*D[y[t],t]-2*k*I*y[t]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
\[
y(t)\to c_2 \left (-\frac {i (2 i-2 i k) (4 i-2 i k) (6 i-2 i k) (8 i-2 i k) t^5}{(20-10 k) (12-8 k) (6-6 k) (2-4 k)}+\frac {i (2 i-2 i k) (4 i-2 i k) (6 i-2 i k) t^4}{(12-8 k) (6-6 k) (2-4 k)}-\frac {i (2 i-2 i k) (4 i-2 i k) t^3}{(6-6 k) (2-4 k)}+\frac {i (2 i-2 i k) t^2}{2-4 k}-i t+1\right )+c_1 \left (\frac {(2 i k-2 i (2 k+1)) (2 i (2 k+2)-2 i k) (2 i (2 k+3)-2 i k) (2 i (2 k+4)-2 i k) (2 i (2 k+5)-2 i k) t^5}{((2 k+1) (2 k+2)-2 k (2 k+2)) ((2 k+2) (2 k+3)-2 k (2 k+3)) ((2 k+3) (2 k+4)-2 k (2 k+4)) ((2 k+4) (2 k+5)-2 k (2 k+5)) ((2 k+5) (2 k+6)-2 k (2 k+6))}-\frac {(2 i k-2 i (2 k+1)) (2 i (2 k+2)-2 i k) (2 i (2 k+3)-2 i k) (2 i (2 k+4)-2 i k) t^4}{((2 k+1) (2 k+2)-2 k (2 k+2)) ((2 k+2) (2 k+3)-2 k (2 k+3)) ((2 k+3) (2 k+4)-2 k (2 k+4)) ((2 k+4) (2 k+5)-2 k (2 k+5))}+\frac {(2 i k-2 i (2 k+1)) (2 i (2 k+2)-2 i k) (2 i (2 k+3)-2 i k) t^3}{((2 k+1) (2 k+2)-2 k (2 k+2)) ((2 k+2) (2 k+3)-2 k (2 k+3)) ((2 k+3) (2 k+4)-2 k (2 k+4))}-\frac {(2 i k-2 i (2 k+1)) (2 i (2 k+2)-2 i k) t^2}{((2 k+1) (2 k+2)-2 k (2 k+2)) ((2 k+2) (2 k+3)-2 k (2 k+3))}+\frac {(2 i k-2 i (2 k+1)) t}{(2 k+1) (2 k+2)-2 k (2 k+2)}+1\right ) t^{2 k+1}
\]
✓ Sympy. Time used: 1.036 (sec). Leaf size: 68
from sympy import *
t = symbols("t")
k = symbols("k")
y = Function("y")
ode = Eq(-2*I*k*y(t) + t*Derivative(y(t), (t, 2)) + (-2*k + 2*I*t)*Derivative(y(t), (t, 2)),0)
ics = {}
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
\[
y{\left (t \right )} = C_{2} \left (- \frac {t^{4}}{24} - \frac {i t^{2}}{2} + 1 + \frac {t^{3} \left (8 - 4 i\right )}{48 k} + \frac {t^{4} \left (4 + 3 i\right )}{48 k^{2}}\right ) + C_{1} t \left (- \frac {i t^{2}}{6} + 1 + \frac {t^{3} \left (2 - i\right )}{24 k}\right ) + O\left (t^{6}\right )
\]