65.13.8 problem 20.4

Internal problem ID [13814]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 20, Series solutions of second order linear equations. Exercises page 195
Problem number : 20.4
Date solved : Tuesday, January 28, 2025 at 06:04:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 32

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

\[ y = \left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1+\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 60

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

\[ y(x)\to c_1 \left (\frac {x^4}{64}+\frac {x^2}{4}+1\right )+c_2 \left (-\frac {3 x^4}{128}-\frac {x^2}{4}+\left (\frac {x^4}{64}+\frac {x^2}{4}+1\right ) \log (x)\right ) \]