13.14.12 problem 12

Internal problem ID [2452]
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 : 12
Date solved : Sunday, March 30, 2025 at 12:01:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 45

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

\[ y = c_1 \sqrt {t}\, \left (1-\frac {1}{2} t +\frac {1}{8} t^{2}-\frac {1}{48} t^{3}+\frac {1}{384} t^{4}-\frac {1}{3840} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 t \left (1-\frac {1}{3} t +\frac {1}{15} t^{2}-\frac {1}{105} t^{3}+\frac {1}{945} t^{4}-\frac {1}{10395} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 86

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

\[ y(t)\to c_1 t \left (-\frac {t^5}{10395}+\frac {t^4}{945}-\frac {t^3}{105}+\frac {t^2}{15}-\frac {t}{3}+1\right )+c_2 \sqrt {t} \left (-\frac {t^5}{3840}+\frac {t^4}{384}-\frac {t^3}{48}+\frac {t^2}{8}-\frac {t}{2}+1\right ) \]

✓ Sympy. Time used: 0.941 (sec). Leaf size: 56

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

\[ y{\left (t \right )} = C_{2} t \left (\frac {t^{4}}{945} - \frac {t^{3}}{105} + \frac {t^{2}}{15} - \frac {t}{3} + 1\right ) + C_{1} \sqrt {t} \left (\frac {t^{4}}{384} - \frac {t^{3}}{48} + \frac {t^{2}}{8} - \frac {t}{2} + 1\right ) + O\left (t^{6}\right ) \]