7.14.19 problem 19

Internal problem ID [2459]
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 : 19
Date solved : Tuesday, September 30, 2025 at 05:36:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 60

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

\[ y = \left (c_1 \,t^{2} \left (1-t +\frac {1}{2} t^{2}-\frac {1}{6} t^{3}+\frac {1}{24} t^{4}-\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (\left (2 t^{2}-2 t^{3}+t^{4}-\frac {1}{3} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \ln \left (t \right )+\left (-2-2 t +3 t^{2}-t^{3}+\frac {1}{9} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right )\right ) t \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 76

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

\[ y(t)\to c_1 \left (\frac {1}{4} t \left (t^4-4 t^2+4 t+4\right )-\frac {1}{2} t^3 \left (t^2-2 t+2\right ) \log (t)\right )+c_2 \left (\frac {t^7}{24}-\frac {t^6}{6}+\frac {t^5}{2}-t^4+t^3\right ) \]

✓ Sympy. Time used: 0.282 (sec). Leaf size: 19

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

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