90.30.10 problem 10

Internal problem ID [25450]
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 517
Problem number : 10
Date solved : Friday, October 03, 2025 at 12:01:38 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-t^{2}+1\right ) y^{\prime \prime }-6 t y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

Order:=6; 
ode:=(-t^2+1)*diff(diff(y(t),t),t)-6*t*diff(y(t),t)-4*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \left (3 t^{4}+2 t^{2}+1\right ) y \left (0\right )+\left (t +\frac {5}{3} t^{3}+\frac {7}{3} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 38

ode=(1-t^2)*D[y[t],{t,2}]-6*t*D[y[t],t]-4*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_2 \left (\frac {7 t^5}{3}+\frac {5 t^3}{3}+t\right )+c_1 \left (3 t^4+2 t^2+1\right ) \]

✓ Sympy. Time used: 0.273 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-6*t*Derivative(y(t), t) + (1 - t**2)*Derivative(y(t), (t, 2)) - 4*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \left (3 t^{4} + 2 t^{2} + 1\right ) + C_{1} t \left (\frac {5 t^{2}}{3} + 1\right ) + O\left (t^{6}\right ) \]