46.5.6 problem 16

Internal problem ID [7342]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. REVIEW QUESTIONS. page 201
Problem number : 16
Date solved : Monday, January 27, 2025 at 02:50:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 x^{3} y^{\prime }+\left (x^{2}-2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 35

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+2*x^3*diff(y(x),x)+(x^2-2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{2} \left (1-\frac {1}{2} x^{2}+\frac {9}{56} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-6 x^{2}+\frac {9}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 44

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+2*x^3*D[y[x],x]+(x^2-2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {3 x^3}{8}-\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {9 x^6}{56}-\frac {x^4}{2}+x^2\right ) \]