7.14.22 problem 22

Internal problem ID [2462]
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 : 22
Date solved : Tuesday, September 30, 2025 at 05:36:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 32

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

\[ y = \left (c_2 \ln \left (t \right )+c_1 \right ) \left (1-t^{2}+\frac {1}{8} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+\left (\frac {5}{4} t^{2}-\frac {9}{32} t^{4}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 56

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

\[ y(t)\to c_1 \left (\frac {t^4}{8}-t^2+1\right )+c_2 \left (-\frac {9 t^4}{32}+\frac {5 t^2}{4}+\left (\frac {t^4}{8}-t^2+1\right ) \log (t)\right ) \]

✓ Sympy. Time used: 0.322 (sec). Leaf size: 17

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

\[ y{\left (t \right )} = C_{1} \left (\frac {t^{4}}{8} - t^{2} + 1\right ) + O\left (t^{6}\right ) \]