28.5.16 problem 9.16

Internal problem ID [4603]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 9. Series Solutions of Differential Equations. Problems at page 426
Problem number : 9.16
Date solved : Monday, January 27, 2025 at 09:26:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 x 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: 0.019 (sec). Leaf size: 33

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

\[ y \left (x \right ) = c_{1} x \left (1+\frac {1}{10} x^{2}+\frac {1}{280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-6 x^{2}-\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 40

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

\[ y(x)\to c_1 \left (-\frac {x^2}{8}+\frac {1}{x^2}-\frac {1}{2}\right )+c_2 \left (\frac {x^5}{280}+\frac {x^3}{10}+x\right ) \]