90.31.19 problem 22

Internal problem ID [25469]
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 : 22
Date solved : Friday, October 03, 2025 at 12:01:51 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t y^{\prime \prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 58

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

\[ y = c_1 t \left (1+2 t +\frac {4}{3} t^{2}+\frac {4}{9} t^{3}+\frac {4}{45} t^{4}+\frac {8}{675} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (\ln \left (t \right ) \left (4 t +8 t^{2}+\frac {16}{3} t^{3}+\frac {16}{9} t^{4}+\frac {16}{45} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (1-12 t^{2}-\frac {112}{9} t^{3}-\frac {140}{27} t^{4}-\frac {808}{675} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 85

ode=t*D[y[t],{t,2}]-4*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \left (\frac {4}{9} t \left (4 t^3+12 t^2+18 t+9\right ) \log (t)+\frac {1}{27} \left (-188 t^4-480 t^3-540 t^2-108 t+27\right )\right )+c_2 \left (\frac {4 t^5}{45}+\frac {4 t^4}{9}+\frac {4 t^3}{3}+2 t^2+t\right ) \]

✓ Sympy. Time used: 0.192 (sec). Leaf size: 34

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

\[ y{\left (t \right )} = C_{1} t \left (\frac {4 t^{4}}{45} + \frac {4 t^{3}}{9} + \frac {4 t^{2}}{3} + 2 t + 1\right ) + O\left (t^{6}\right ) \]