66.2.15 problem Problem 15

Internal problem ID [13843]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 15
Date solved : Monday, March 31, 2025 at 08:14:56 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 28

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

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

✓ Sympy. Time used: 0.632 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*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)
 

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