54.9.16 problem 16

Internal problem ID [8686]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 16
Date solved : Monday, January 27, 2025 at 04:19:14 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 37

Order:=8; 
dsolve((1+x^2)*diff(y(x),x$2)-2*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (x^{2}+1\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}-\frac {1}{15} x^{5}+\frac {1}{35} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 38

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

\[ y(x)\to c_1 \left (x^2+1\right )+c_2 \left (\frac {x^7}{35}-\frac {x^5}{15}+\frac {x^3}{3}+x\right ) \]