36.6.11 problem 11

Internal problem ID [6372]
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 : 11
Date solved : Monday, January 27, 2025 at 01:58:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 76

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

\[ y = \left (1-\frac {\left (x -2\right )^{2}}{8}+\frac {\left (x -2\right )^{3}}{32}-\frac {3 \left (x -2\right )^{4}}{512}+\frac {\left (x -2\right )^{5}}{2048}\right ) y \left (2\right )+\left (x -2+\frac {\left (x -2\right )^{2}}{8}-\frac {7 \left (x -2\right )^{3}}{96}+\frac {37 \left (x -2\right )^{4}}{1536}-\frac {211 \left (x -2\right )^{5}}{30720}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (\frac {(x-2)^5}{2048}-\frac {3}{512} (x-2)^4+\frac {1}{32} (x-2)^3-\frac {1}{8} (x-2)^2+1\right )+c_2 \left (-\frac {211 (x-2)^5}{30720}+\frac {37 (x-2)^4}{1536}-\frac {7}{96} (x-2)^3+\frac {1}{8} (x-2)^2+x-2\right ) \]