54.5.12 problem 11 (solved as series)

Internal problem ID [8627]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 11 (solved as series)
Date solved : Monday, January 27, 2025 at 04:17:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\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: 36

Order:=8; 
dsolve(x*diff(y(x),x$2)+diff(y(x),x)-x*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\frac {1}{2304} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {3}{128} x^{4}-\frac {11}{13824} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 81

AsymptoticDSolveValue[x*D[y[x],{x,2}]+D[y[x],x]-x*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^6}{2304}+\frac {x^4}{64}+\frac {x^2}{4}+1\right )+c_2 \left (-\frac {11 x^6}{13824}-\frac {3 x^4}{128}-\frac {x^2}{4}+\left (\frac {x^6}{2304}+\frac {x^4}{64}+\frac {x^2}{4}+1\right ) \log (x)\right ) \]