20.26.26 problem 20

Internal problem ID [4051]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 20
Date solved : Monday, January 27, 2025 at 08:07:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x^{2} y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 45

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

\[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {1}{2} x +\frac {3}{20} x^{2}+\frac {1}{30} x^{3}+\frac {1}{168} x^{4}+\frac {1}{1120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12+6 x -x^{3}-\frac {1}{2} x^{4}-\frac {3}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (-\frac {x^3}{24}-\frac {x^2}{12}+\frac {1}{x}+\frac {1}{2}\right )+c_2 \left (\frac {x^6}{168}+\frac {x^5}{30}+\frac {3 x^4}{20}+\frac {x^3}{2}+x^2\right ) \]