13.14.8 problem 8

Internal problem ID [2448]
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 : 8
Date solved : Sunday, March 30, 2025 at 12:01:45 AM
CAS classification : [_Laguerre]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 44

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

\[ y = c_1 \sqrt {t}\, \left (1+\frac {2}{3} t +\frac {4}{15} t^{2}+\frac {8}{105} t^{3}+\frac {16}{945} t^{4}+\frac {32}{10395} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1+t +\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{24} t^{4}+\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 81

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

\[ y(t)\to c_1 \sqrt {t} \left (\frac {32 t^5}{10395}+\frac {16 t^4}{945}+\frac {8 t^3}{105}+\frac {4 t^2}{15}+\frac {2 t}{3}+1\right )+c_2 \left (\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2}+t+1\right ) \]

✓ Sympy. Time used: 1.119 (sec). Leaf size: 65

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{5}}{120} + \frac {t^{4}}{24} + \frac {t^{3}}{6} + \frac {t^{2}}{2} + t + 1\right ) + C_{1} \sqrt {t} \left (\frac {16 t^{4}}{945} + \frac {8 t^{3}}{105} + \frac {4 t^{2}}{15} + \frac {2 t}{3} + 1\right ) + O\left (t^{6}\right ) \]