90.31.16 problem 19

Internal problem ID [25466]
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 : 19
Date solved : Friday, October 03, 2025 at 12:01:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 45

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

\[ y = c_1 \,t^{2} \left (1-\frac {1}{2} t +\frac {3}{20} t^{2}-\frac {1}{30} t^{3}+\frac {1}{168} t^{4}-\frac {1}{1120} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\frac {c_2 \left (12-6 t +t^{3}-\frac {1}{2} t^{4}+\frac {3}{20} t^{5}+\operatorname {O}\left (t^{6}\right )\right )}{t} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 63

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

\[ y(t)\to c_1 \left (-\frac {t^3}{24}+\frac {t^2}{12}+\frac {1}{t}-\frac {1}{2}\right )+c_2 \left (\frac {t^6}{168}-\frac {t^5}{30}+\frac {3 t^4}{20}-\frac {t^3}{2}+t^2\right ) \]

✓ Sympy. Time used: 0.280 (sec). Leaf size: 36

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

\[ y{\left (t \right )} = C_{2} t^{2} \left (- \frac {t^{3}}{30} + \frac {3 t^{2}}{20} - \frac {t}{2} + 1\right ) + \frac {C_{1} \left (\frac {t}{2} - 1\right )}{t} + O\left (t^{6}\right ) \]