90.31.13 problem 16

Internal problem ID [25463]
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 : 16
Date solved : Friday, October 03, 2025 at 12:01:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 45

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

\[ y = c_1 \sqrt {t}\, \left (1-t +\frac {1}{6} t^{2}-\frac {1}{90} t^{3}+\frac {1}{2520} t^{4}-\frac {1}{113400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 t \left (1-\frac {1}{3} t +\frac {1}{30} t^{2}-\frac {1}{630} t^{3}+\frac {1}{22680} t^{4}-\frac {1}{1247400} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 84

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

\[ y(t)\to c_1 t \left (-\frac {t^5}{1247400}+\frac {t^4}{22680}-\frac {t^3}{630}+\frac {t^2}{30}-\frac {t}{3}+1\right )+c_2 \sqrt {t} \left (-\frac {t^5}{113400}+\frac {t^4}{2520}-\frac {t^3}{90}+\frac {t^2}{6}-t+1\right ) \]

✓ Sympy. Time used: 0.324 (sec). Leaf size: 54

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

\[ y{\left (t \right )} = C_{2} t \left (\frac {t^{4}}{22680} - \frac {t^{3}}{630} + \frac {t^{2}}{30} - \frac {t}{3} + 1\right ) + C_{1} \sqrt {t} \left (\frac {t^{4}}{2520} - \frac {t^{3}}{90} + \frac {t^{2}}{6} - t + 1\right ) + O\left (t^{6}\right ) \]