50.23.1 problem 1(a)

Internal problem ID [8164]
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. Problems for review and discovert. (B) Challenge Problems . Page 194
Problem number : 1(a)
Date solved : Monday, January 27, 2025 at 03:45:04 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} \infty \end{align*}

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 146

Order:=8; 
dsolve(x^3*diff(y(x),x$2)+x^2*diff(y(x),x)+y(x)=0,y(x),type='series',x=Infinity);
 

\[ y = \frac {1778112000 c_{2} \left (O\left (\frac {1}{x^{8}}\right ) x^{7}+x^{7}-x^{6}+\frac {x^{5}}{4}-\frac {x^{4}}{36}+\frac {x^{3}}{576}-\frac {x^{2}}{14400}+\frac {x}{518400}-\frac {1}{25401600}\right ) \ln \left (\frac {1}{x}\right )[phone] x^{7} \left (c_{1} +c_{2} \right ) O\left (\frac {1}{x^{8}}\right )[phone] c_{1} x^{7}+\left (-1778112000 c_{1} [phone] c_{2} \right ) x^{6}+\left (444528000 c_{1} -1333584000 c_{2} \right ) x^{5}+\left (-49392000 c_{1} [phone] c_{2} \right ) x^{4}+\left (3087000 c_{1} -12862500 c_{2} \right ) x^{3}+\left (-123480 c_{1} +563892 c_{2} \right ) x^{2}+\left (3430 c_{1} -16807 c_{2} \right ) x -70 c_{1} +363 c_{2}}{1778112000 x^{7}} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 171

AsymptoticDSolveValue[x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+y[x]==0,y[x],{x,Infinity,"8"-1}]
 

\[ y(x)\to c_1 \left (-\frac {1}{25401600 x^7}+\frac {1}{518400 x^6}-\frac {1}{14400 x^5}+\frac {1}{576 x^4}-\frac {1}{36 x^3}+\frac {1}{4 x^2}-\frac {1}{x}+1\right )+c_2 \left (\frac {121}{592704000 x^7}+\frac {\log (x)}{25401600 x^7}-\frac {49}{5184000 x^6}-\frac {\log (x)}{518400 x^6}+\frac {137}{432000 x^5}+\frac {\log (x)}{14400 x^5}-\frac {25}{3456 x^4}-\frac {\log (x)}{576 x^4}+\frac {11}{108 x^3}+\frac {\log (x)}{36 x^3}-\frac {3}{4 x^2}-\frac {\log (x)}{4 x^2}+\frac {2}{x}+\frac {\log (x)}{x}-\log (x)\right ) \]