90.31.8 problem 8

Internal problem ID [25458]
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 : 8
Date solved : Friday, October 03, 2025 at 12:01:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} y^{\prime \prime }+t \,{\mathrm e}^{t} y^{\prime }+4 \left (1-4 t \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 69

Order:=6; 
ode:=t^2*diff(diff(y(t),t),t)+t*exp(t)*diff(y(t),t)+4*(1-4*t)*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = c_1 \,t^{-2 i} \left (1+\left (\frac {8}{17}+\frac {66 i}{17}\right ) t +\left (-\frac {1029}{170}+\frac {999 i}{340}\right ) t^{2}+\left (-\frac {40369}{7650}-\frac {10039 i}{2550}\right ) t^{3}+\left (\frac {185837}{489600}-\frac {663593 i}{163200}\right ) t^{4}+\left (\frac {8927289}{5576000}-\frac {12371159 i}{16728000}\right ) t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \,t^{2 i} \left (1+\left (\frac {8}{17}-\frac {66 i}{17}\right ) t +\left (-\frac {1029}{170}-\frac {999 i}{340}\right ) t^{2}+\left (-\frac {40369}{7650}+\frac {10039 i}{2550}\right ) t^{3}+\left (\frac {185837}{489600}+\frac {663593 i}{163200}\right ) t^{4}+\left (\frac {8927289}{5576000}+\frac {12371159 i}{16728000}\right ) t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 94

ode=t^2*D[y[t],{t,2}]+t*Exp[t]*D[y[t],t]+4*(1-4*t)*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to \left (\frac {11}{489600}+\frac {7 i}{489600}\right ) c_1 t^{2 i} \left ((93998+121163 i) t^4-(87808-231104 i) t^3-(250992-28944 i) t^2-(63360+132480 i) t+(31680-20160 i)\right )-\left (\frac {7}{489600}+\frac {11 i}{489600}\right ) c_2 t^{-2 i} \left ((121163+93998 i) t^4+(231104-87808 i) t^3+(28944-250992 i) t^2-(132480+63360 i) t-(20160-31680 i)\right ) \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + t*exp(t)*Derivative(y(t), t) + (4 - 16*t)*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None