90.31.10 problem 10

Internal problem ID [25460]
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 : 10
Date solved : Friday, October 03, 2025 at 12:01:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} y^{\prime \prime }+3 t \left (1+3 t \right ) y^{\prime }+\left (-t^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 48

Order:=6; 
ode:=t^2*diff(diff(y(t),t),t)+3*t*(1+3*t)*diff(y(t),t)+(-t^2+1)*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \frac {\left (c_2 \ln \left (t \right )+c_1 \right ) \left (1+9 t +\frac {1}{4} t^{2}+\frac {3}{4} t^{3}-\frac {53}{64} t^{4}+\frac {1479}{1600} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (\left (-27\right ) t -\frac {41}{2} t^{2}+\frac {67}{4} t^{3}-\frac {2577}{128} t^{4}+\frac {357469}{16000} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2}{t} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 122

ode=t^2*D[y[t],{t,2}]+3*t*(1+3*t)*D[y[t],t]+(1-t^2)*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to \frac {c_1 \left (\frac {1479 t^5}{1600}-\frac {53 t^4}{64}+\frac {3 t^3}{4}+\frac {t^2}{4}+9 t+1\right )}{t}+c_2 \left (\frac {\frac {357469 t^5}{16000}-\frac {2577 t^4}{128}+\frac {67 t^3}{4}-\frac {41 t^2}{2}-27 t}{t}+\frac {\left (\frac {1479 t^5}{1600}-\frac {53 t^4}{64}+\frac {3 t^3}{4}+\frac {t^2}{4}+9 t+1\right ) \log (t)}{t}\right ) \]

✓ Sympy. Time used: 0.357 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + 3*t*(3*t + 1)*Derivative(y(t), t) + (1 - t**2)*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (t \right )} = \frac {C_{1} \left (- \frac {54569 t^{6}}{57600} + \frac {1479 t^{5}}{1600} - \frac {53 t^{4}}{64} + \frac {3 t^{3}}{4} + \frac {t^{2}}{4} + 9 t + 1\right )}{t} + O\left (t^{6}\right ) \]