45.3.15 problem 17

Internal problem ID [7273]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.3.1 page 250
Problem number : 17
Date solved : Monday, January 27, 2025 at 02:49:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \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)+(x^2-2)*y(x)=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 44

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

\[ y(x)\to c_1 \left (-\frac {x^3}{8}+\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {x^6}{280}-\frac {x^4}{10}+x^2\right ) \]