90.31.17 problem 20

Internal problem ID [25467]
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 : 20
Date solved : Friday, October 03, 2025 at 12:01:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 44

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

\[ y = c_1 \left (1+\frac {1}{6} a \,t^{2}+\frac {1}{120} a^{2} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+\frac {c_2 \left (1+\frac {1}{2} a \,t^{2}+\frac {1}{24} a^{2} t^{4}+\operatorname {O}\left (t^{6}\right )\right )}{t} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 50

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

\[ y(t)\to c_1 \left (\frac {a^2 t^3}{24}+\frac {a t}{2}+\frac {1}{t}\right )+c_2 \left (\frac {a^2 t^4}{120}+\frac {a t^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.375 (sec). Leaf size: 53

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

\[ y{\left (t \right )} = C_{2} \left (\frac {a^{2} t^{4}}{120} + \frac {a t^{2}}{6} + 1\right ) + \frac {C_{1} \left (\frac {a^{3} t^{6}}{720} + \frac {a^{2} t^{4}}{24} + \frac {a t^{2}}{2} + 1\right )}{t} + O\left (t^{6}\right ) \]