90.31.12 problem 15

Internal problem ID [25462]
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 : 15
Date solved : Friday, October 03, 2025 at 12:01:46 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 32

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

\[ y = c_1 \,t^{3} \left (1-\frac {1}{10} t^{2}+\frac {1}{280} t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (12+6 t^{2}-\frac {3}{2} t^{4}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 66

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

\[ y(t)\to c_1 \left (-\frac {t^5}{30}-\frac {t^4}{8}-\frac {t^3}{3}-\frac {t^2}{2}+1\right )+c_2 \left (\frac {t^5}{24}+\frac {t^4}{6}+\frac {t^3}{2}+t^2+t\right ) \]

✓ Sympy. Time used: 0.255 (sec). Leaf size: 41

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

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