90.31.6 problem 6

Internal problem ID [25456]
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 537
Problem number : 6
Date solved : Friday, October 03, 2025 at 12:01:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 t y^{\prime \prime }+y^{\prime }+y t&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 32

Order:=6; 
ode:=2*t*diff(diff(y(t),t),t)+diff(y(t),t)+t*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = c_1 \sqrt {t}\, \left (1-\frac {1}{10} t^{2}+\frac {1}{360} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1-\frac {1}{6} t^{2}+\frac {1}{168} t^{4}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 47

ode=2*t*D[y[t],{t,2}]+D[y[t],t]+t*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \sqrt {t} \left (\frac {t^4}{360}-\frac {t^2}{10}+1\right )+c_2 \left (\frac {t^4}{168}-\frac {t^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.267 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*y(t) + 2*t*Derivative(y(t), (t, 2)) + Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{4}}{168} - \frac {t^{2}}{6} + 1\right ) + C_{1} \sqrt {t} \left (\frac {t^{4}}{360} - \frac {t^{2}}{10} + 1\right ) + O\left (t^{6}\right ) \]