82.2.2 problem 24-2

Internal problem ID [21763]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 24. Power series about an ordinary point. Page 719
Problem number : 24-2
Date solved : Thursday, October 02, 2025 at 08:01:54 PM
CAS classification : [[_2nd_order, _missing_x]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

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

\[ y = \left (1-2 x^{2}+\frac {2}{3} x^{4}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {2}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

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

\[ y(x)\to c_2 \left (\frac {2 x^5}{15}-\frac {2 x^3}{3}+x\right )+c_1 \left (\frac {2 x^4}{3}-2 x^2+1\right ) \]

✓ Sympy. Time used: 0.171 (sec). Leaf size: 32

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

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