7.14.13 problem 13

Internal problem ID [2453]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number : 13
Date solved : Tuesday, September 30, 2025 at 05:36:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{3} y^{\prime \prime }-t y^{\prime }-\left (t^{2}+\frac {5}{4}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=t^3*diff(diff(y(t),t),t)-t*diff(y(t),t)-(t^2+5/4)*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 97

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

\[ y(t)\to c_2 e^{-1/t} \left (-\frac {239684276027 t^5}{8388608}+\frac {1648577803 t^4}{524288}-\frac {3127415 t^3}{8192}+\frac {26113 t^2}{512}-\frac {117 t}{16}+1\right ) t^{13/4}+\frac {c_1 \left (-\frac {784957 t^5}{8388608}-\frac {152693 t^4}{524288}-\frac {7649 t^3}{8192}-\frac {31 t^2}{512}+\frac {45 t}{16}+1\right )}{t^{5/4}} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**3*Derivative(y(t), (t, 2)) - t*Derivative(y(t), t) - (t**2 + 5/4)*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE t**3*Derivative(y(t), (t, 2)) - t*Derivative(y(t), t) - (t**2 + 5/4)*y(t) does not match hint 2nd_power_series_regular