50.20.4 problem 3(b)

Internal problem ID [8132]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.5. More on Regular Singular Points. Page 183
Problem number : 3(b)
Date solved : Monday, January 27, 2025 at 03:44:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x^{2} 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: 0.017 (sec). Leaf size: 55

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

\[ y = c_{1} x^{2} \left (1+\frac {1}{2} x +\frac {1}{20} x^{2}-\frac {1}{60} x^{3}-\frac {1}{210} x^{4}-\frac {1}{3360} x^{5}+\frac {1}{20160} x^{6}+\frac {1}{100800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12+6 x +6 x^{2}+5 x^{3}+x^{4}-\frac {1}{5} x^{5}-\frac {1}{10} x^{6}-\frac {3}{280} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.068 (sec). Leaf size: 96

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

\[ y(x)\to c_1 \left (-\frac {x^5}{120}-\frac {x^4}{60}+\frac {x^3}{12}+\frac {5 x^2}{12}+\frac {x}{2}+\frac {1}{x}+\frac {1}{2}\right )+c_2 \left (\frac {x^8}{20160}-\frac {x^7}{3360}-\frac {x^6}{210}-\frac {x^5}{60}+\frac {x^4}{20}+\frac {x^3}{2}+x^2\right ) \]