80.9.7 problem 7

Internal problem ID [21389]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 9. Solutions by infinite series and Bessel functions. Excercise 10.6 at page 223
Problem number : 7
Date solved : Thursday, October 02, 2025 at 07:30:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} x^{\prime \prime }-3 x^{\prime } t +\left (4-t \right ) x&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 48

Order:=6; 
ode:=t^2*diff(diff(x(t),t),t)-3*t*diff(x(t),t)+(4-t)*x(t) = 0; 
dsolve(ode,x(t),type='series',t=0);
 

\[ x = t^{2} \left (\left (c_2 \ln \left (t \right )+c_1 \right ) \left (1+t +\frac {1}{4} t^{2}+\frac {1}{36} t^{3}+\frac {1}{576} t^{4}+\frac {1}{14400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (\left (-2\right ) t -\frac {3}{4} t^{2}-\frac {11}{108} t^{3}-\frac {25}{3456} t^{4}-\frac {137}{432000} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 1182

ode=t^2*D[x[t],{t,2}]+3*t*D[x[t],t]+(4-t)*x[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},x[t],{t,0,5}]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t**2*Derivative(x(t), (t, 2)) + 3*t*Derivative(x(t), t) + (4 - t)*x(t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None