50.18.12 problem 6

Internal problem ID [8106]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.3. Second-Order Linear Equations: Ordinary Points. Page 169
Problem number : 6
Date solved : Sunday, March 30, 2025 at 12:45:31 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+x 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:=8; 
ode:=diff(diff(y(x),x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}+\frac {1}{504} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

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

ode=D[y[x],{x,2}]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {x^7}{504}-\frac {x^4}{12}+x\right )+c_1 \left (\frac {x^6}{180}-\frac {x^3}{6}+1\right ) \]

✓ Sympy. Time used: 0.716 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(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=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6}}{180} - \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (\frac {x^{6}}{504} - \frac {x^{3}}{12} + 1\right ) + O\left (x^{8}\right ) \]