7.14.21 problem 21

Internal problem ID [2461]
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 : 21
Date solved : Tuesday, September 30, 2025 at 05:36:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t y^{\prime \prime }+t y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 60

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

\[ y = c_1 t \left (1-\frac {3}{2} t +t^{2}-\frac {5}{12} t^{3}+\frac {1}{8} t^{4}-\frac {7}{240} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (\ln \left (t \right ) \left (\left (-2\right ) t +3 t^{2}-2 t^{3}+\frac {5}{6} t^{4}-\frac {1}{4} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (1-t -2 t^{2}+\frac {5}{2} t^{3}-\frac {49}{36} t^{4}+\frac {23}{48} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 83

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

\[ y(t)\to c_1 \left (\frac {1}{6} t \left (5 t^3-12 t^2+18 t-12\right ) \log (t)+\frac {1}{36} \left (-79 t^4+162 t^3-180 t^2+36 t+36\right )\right )+c_2 \left (\frac {t^5}{8}-\frac {5 t^4}{12}+t^3-\frac {3 t^2}{2}+t\right ) \]

✓ Sympy. Time used: 0.228 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = C_{1} t \left (\frac {t^{4}}{8} - \frac {5 t^{3}}{12} + t^{2} - \frac {3 t}{2} + 1\right ) + O\left (t^{6}\right ) \]