46.1.8 problem 13

Internal problem ID [7298]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.1. page 174
Problem number : 13
Date solved : Monday, January 27, 2025 at 02:49:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (x^{2}+1\right ) 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: 39

Order:=6; 
dsolve(diff(y(x),x$2)+(1+x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{24} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{24} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[D[y[x],{x,2}]+(1+x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (-\frac {x^5}{24}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {x^4}{24}-\frac {x^2}{2}+1\right ) \]