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 : Monday, January 27, 2025 at 03:43:51 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 39

Order:=8; 
dsolve(diff(y(x),x$2)+x*y(x)=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[D[y[x],{x,2}]+x*y[x]==0,y[x],{x,0,"8"-1}]
 

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