14.14.22 problem 22

Internal problem ID [2659]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8.2, Regular singular points, the method of Frobenius. Excercises page 216
Problem number : 22
Date solved : Sunday, March 30, 2025 at 12:12:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} 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

Order:=6; 
ode:=t^2*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);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 77

ode=t^2*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}]
 

\[ y(t)\to c_2 e^{\frac {1}{t}} \left (\frac {39467 t^5}{24}+\frac {2135 t^4}{8}+\frac {305 t^3}{6}+\frac {23 t^2}{2}+3 t+1\right ) t^2+c_1 \left (\frac {32 t^5}{15}-t^4+\frac {4 t^3}{3}-2 t^2+1\right ) \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + 4*t*y(t) + (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)
 

ValueError : ODE t**2*Derivative(y(t), (t, 2)) + 4*t*y(t) + (1 - t**2)*Derivative(y(t), t) does not match hint 2nd_power_series_regular