67.25.14 problem 35.3 (h)

Internal problem ID [17009]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.3 (h)
Date solved : Thursday, October 02, 2025 at 01:41:50 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (x^{2}+4\right )^{2} y^{\prime \prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 39

Order:=6; 
ode:=(x^2+4)^2*diff(diff(y(x),x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{32} x^{2}+\frac {17}{6144} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{96} x^{3}+\frac {49}{30720} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (\frac {49 x^5}{30720}-\frac {x^3}{96}+x\right )+c_1 \left (\frac {17 x^4}{6144}-\frac {x^2}{32}+1\right ) \]

✓ Sympy. Time used: 0.286 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = - \frac {49 x^{5} r{\left (3 \right )}}{320} + C_{2} \left (\frac {17 x^{4}}{6144} - \frac {x^{2}}{32} + 1\right ) + C_{1} x + O\left (x^{6}\right ) \]