Internal problem ID [5743]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime }+y^{\prime } x^{2}+y=0} \end {gather*} With the expansion point for the power series method at \(x = \infty \).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 207
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);
\[ y \relax (x ) = \left (1-\frac {\left (x -\mathit {Infinity} \right )^{2}}{2 \mathit {Infinity}^{3}}+\frac {2 \left (x -\mathit {Infinity} \right )^{3}}{3 \mathit {Infinity}^{4}}+\frac {\left (-18 \mathit {Infinity} +1\right ) \left (x -\mathit {Infinity} \right )^{4}}{24 \mathit {Infinity}^{6}}+\frac {\left (96 \mathit {Infinity} -14\right ) \left (x -\mathit {Infinity} \right )^{5}}{120 \mathit {Infinity}^{7}}+\frac {\left (-600 \mathit {Infinity}^{2}+156 \mathit {Infinity} -1\right ) \left (x -\mathit {Infinity} \right )^{6}}{720 \mathit {Infinity}^{9}}+\frac {\left (4320 \mathit {Infinity}^{2}-1692 \mathit {Infinity} +30\right ) \left (x -\mathit {Infinity} \right )^{7}}{5040 \mathit {Infinity}^{10}}\right ) y \left (\mathit {Infinity} \right )+\left (x -\mathit {Infinity} -\frac {\left (x -\mathit {Infinity} \right )^{2}}{2 \mathit {Infinity}}+\frac {\left (2 \mathit {Infinity}^{2}-\mathit {Infinity} \right ) \left (x -\mathit {Infinity} \right )^{3}}{6 \mathit {Infinity}^{4}}+\frac {\left (-6 \mathit {Infinity}^{3}+8 \mathit {Infinity}^{2}\right ) \left (x -\mathit {Infinity} \right )^{4}}{24 \mathit {Infinity}^{6}}+\frac {\left (24 \mathit {Infinity}^{3}-58 \mathit {Infinity}^{2}+\mathit {Infinity} \right ) \left (x -\mathit {Infinity} \right )^{5}}{120 \mathit {Infinity}^{7}}+\frac {\left (-120 \mathit {Infinity}^{4}+444 \mathit {Infinity}^{3}-21 \mathit {Infinity}^{2}\right ) \left (x -\mathit {Infinity} \right )^{6}}{720 \mathit {Infinity}^{9}}+\frac {\left (720 \mathit {Infinity}^{4}-3708 \mathit {Infinity}^{3}+324 \mathit {Infinity}^{2}-\mathit {Infinity} \right ) \left (x -\mathit {Infinity} \right )^{7}}{5040 \mathit {Infinity}^{10}}\right ) D\relax (y )\left (\mathit {Infinity} \right )+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 171
AsymptoticDSolveValue[x^3*y''[x]+x^2*y'[x]+y[x]==0,y[x],{x,Infinity,7}]
\[ y(x)\to c_1 \left (-\frac {1}{25401600 x^7}+\frac {1}{518400 x^6}-\frac {1}{14400 x^5}+\frac {1}{576 x^4}-\frac {1}{36 x^3}+\frac {1}{4 x^2}-\frac {1}{x}+1\right )+c_2 \left (\frac {121}{592704000 x^7}+\frac {\log (x)}{25401600 x^7}-\frac {49}{5184000 x^6}-\frac {\log (x)}{518400 x^6}+\frac {137}{432000 x^5}+\frac {\log (x)}{14400 x^5}-\frac {25}{3456 x^4}-\frac {\log (x)}{576 x^4}+\frac {11}{108 x^3}+\frac {\log (x)}{36 x^3}-\frac {3}{4 x^2}-\frac {\log (x)}{4 x^2}+\frac {2}{x}+\frac {\log (x)}{x}-\log (x)\right ) \]