36.6.10 problem 10

Internal problem ID [6371]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 10
Date solved : Monday, January 27, 2025 at 01:58:21 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 69

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

\[ y = \left (1-\frac {\left (x -2\right )^{2}}{4}+\frac {\left (x -2\right )^{3}}{24}-\frac {\left (x -2\right )^{4}}{192}\right ) y \left (2\right )+\left (x -2+\frac {\left (x -2\right )^{2}}{4}-\frac {\left (x -2\right )^{3}}{12}+\frac {\left (x -2\right )^{4}}{48}-\frac {\left (x -2\right )^{5}}{192}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 78

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

\[ y(x)\to c_1 \left (-\frac {1}{192} (x-2)^4+\frac {1}{24} (x-2)^3-\frac {1}{4} (x-2)^2+1\right )+c_2 \left (-\frac {1}{192} (x-2)^5+\frac {1}{48} (x-2)^4-\frac {1}{12} (x-2)^3+\frac {1}{4} (x-2)^2+x-2\right ) \]