14.14.9 problem 9

Internal problem ID [2646]
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 : 9
Date solved : Tuesday, March 04, 2025 at 02:32:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 38

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

\[ y = c_1 \sqrt {t}\, \left (1+\frac {1}{2} t +\frac {1}{40} t^{2}-\frac {1}{1680} t^{3}+\frac {1}{40320} t^{4}-\frac {1}{887040} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1+2 t +\frac {1}{3} t^{2}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 62

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

\[ y(t)\to c_2 \left (\frac {t^2}{3}+2 t+1\right )+c_1 \sqrt {t} \left (-\frac {t^5}{887040}+\frac {t^4}{40320}-\frac {t^3}{1680}+\frac {t^2}{40}+\frac {t}{2}+1\right ) \]

✓ Sympy. Time used: 0.895 (sec). Leaf size: 44

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{2}}{3} + 2 t + 1\right ) + C_{1} \sqrt {t} \left (\frac {t^{4}}{40320} - \frac {t^{3}}{1680} + \frac {t^{2}}{40} + \frac {t}{2} + 1\right ) + O\left (t^{6}\right ) \]