50.19.18 problem 5

Internal problem ID [8126]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 5
Date solved : Monday, January 27, 2025 at 03:44:19 PM
CAS classification : [_Lienard]

\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: 0.011 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 81

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

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