90.31.18 problem 21

Internal problem ID [25468]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 7. Power series methods. Exercises at page 537
Problem number : 21
Date solved : Friday, October 03, 2025 at 12:01:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} t y^{\prime \prime }+\left (t -1\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: 44

Order:=6; 
ode:=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);
 

\[ y = c_1 \,t^{2} \left (1+\operatorname {O}\left (t^{6}\right )\right )+\left (2 t^{2}+\operatorname {O}\left (t^{6}\right )\right ) \ln \left (t \right ) c_2 +\left (-2+4 t -3 t^{2}-\frac {2}{3} t^{3}+\frac {1}{12} t^{4}-\frac {1}{90} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 45

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

\[ y(t)\to c_2 t^2+c_1 \left (\frac {1}{24} \left (-t^4+8 t^3+48 t^2-48 t+24\right )-t^2 \log (t)\right ) \]

✓ Sympy. Time used: 0.236 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(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)
 

\[ y{\left (t \right )} = C_{1} t^{2} + O\left (t^{6}\right ) \]