10.13.21 problem 27

Internal problem ID [1381]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.2, Series Solutions Near an Ordinary Point, Part I. page 263
Problem number : 27
Date solved : Saturday, March 29, 2025 at 10:53:47 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+x^{2} y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 12

Order:=6; 
ode:=diff(diff(y(x),x),x)+x^2*y(x) = 0; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = 1-\frac {1}{12} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 12

ode=D[y[x],{x,2}]+x^2*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to 1-\frac {x^4}{12} \]

✓ Sympy. Time used: 0.694 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {x^{4}}{12}\right ) + C_{1} x \left (1 - \frac {x^{4}}{20}\right ) + O\left (x^{6}\right ) \]