54.3.13 problem 13

Internal problem ID [8569]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 13
Date solved : Monday, January 27, 2025 at 04:16:33 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*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 29

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

\[ y = \left (1-\frac {x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 28

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

\[ y(x)\to c_2 \left (x-\frac {x^5}{20}\right )+c_1 \left (1-\frac {x^4}{12}\right ) \]