Internal problem ID [5468]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 26. Integration in series (singular points). Supplemetary problems. Page
218
Problem number: 20.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {2 x^{3} y^{\prime \prime }+y^{\prime } x^{2}+y=0} \] With the expansion point for the power series method at \(x = \infty \).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 117
Order:=6; dsolve(2*x^3*diff(y(x),x$2)+x^2*diff(y(x),x)+y(x)=0,y(x),type='series',x=Infinity);
\[ y = \left (1-\frac {\left (x -\operatorname {Infinity} \right )^{2}}{4 \operatorname {Infinity}^{3}}+\frac {7 \left (x -\operatorname {Infinity} \right )^{3}}{24 \operatorname {Infinity}^{4}}+\frac {\left (-59 \operatorname {Infinity} +2\right ) \left (x -\operatorname {Infinity} \right )^{4}}{192 \operatorname {Infinity}^{6}}+\frac {\left (605 \operatorname {Infinity} -52\right ) \left (x -\operatorname {Infinity} \right )^{5}}{1920 \operatorname {Infinity}^{7}}\right ) y \left (\operatorname {Infinity} \right )+\left (x -\operatorname {Infinity} -\frac {\left (x -\operatorname {Infinity} \right )^{2}}{4 \operatorname {Infinity}}+\frac {\left (3 \operatorname {Infinity}^{2}-2 \operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{3}}{24 \operatorname {Infinity}^{4}}-\frac {5 \left (\operatorname {Infinity} -\frac {28}{15}\right ) \left (x -\operatorname {Infinity} \right )^{4}}{64 \operatorname {Infinity}^{4}}+\frac {\left (105 \operatorname {Infinity}^{3}-370 \operatorname {Infinity}^{2}+4 \operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{5}}{1920 \operatorname {Infinity}^{7}}\right ) y^{\prime }\left (\operatorname {Infinity} \right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 96
AsymptoticDSolveValue[2*x^3*y''[x]+x^2*y'[x]+y[x]==0,y[x],{x,Infinity,5}]
\[ y(x)\to c_2 \left (\frac {1}{6 x^{3/2}}-\frac {1}{90 x^{5/2}}+\frac {1}{2520 x^{7/2}}-\frac {1}{113400 x^{9/2}}+\sqrt {x}-\frac {1}{\sqrt {x}}\right )+c_1 \left (-\frac {1}{1247400 x^5}+\frac {1}{22680 x^4}-\frac {1}{630 x^3}+\frac {1}{30 x^2}-\frac {1}{3 x}+1\right ) \]