20.27.2 problem 3

Internal problem ID [4062]
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 11.6. page 783
Problem number : 3
Date solved : Monday, January 27, 2025 at 08:07:20 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 42

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

\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 59

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

\[ y(x)\to c_1 \left (\frac {1}{16} \left (x^2-8\right ) x^2 \log (x)+\frac {1}{64} \left (-5 x^4+16 x^2+64\right )\right )+c_2 \left (\frac {x^6}{192}-\frac {x^4}{8}+x^2\right ) \]