Internal problem ID [5469]
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: 21.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{3} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = \infty \).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 179
Order:=6; dsolve(x^3*diff(y(x),x$2)+(x^2+x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=Infinity);
\[ y = \left (1+\frac {\left (x -\operatorname {Infinity} \right )^{2}}{2 \operatorname {Infinity}^{3}}+\frac {\left (-4 \operatorname {Infinity} -1\right ) \left (x -\operatorname {Infinity} \right )^{3}}{6 \operatorname {Infinity}^{5}}+\frac {\left (18 \operatorname {Infinity}^{2}+10 \operatorname {Infinity} +1\right ) \left (x -\operatorname {Infinity} \right )^{4}}{24 \operatorname {Infinity}^{7}}+\frac {\left (-96 \operatorname {Infinity}^{3}-86 \operatorname {Infinity}^{2}-18 \operatorname {Infinity} -1\right ) \left (x -\operatorname {Infinity} \right )^{5}}{120 \operatorname {Infinity}^{9}}\right ) y \left (\operatorname {Infinity} \right )+\left (x -\operatorname {Infinity} +\frac {\left (-\operatorname {Infinity}^{2}-\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{2}}{2 \operatorname {Infinity}^{3}}+\frac {\left (2 \operatorname {Infinity}^{3}+5 \operatorname {Infinity}^{2}+\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{3}}{6 \operatorname {Infinity}^{5}}+\frac {\left (-6 \operatorname {Infinity}^{4}-26 \operatorname {Infinity}^{3}-11 \operatorname {Infinity}^{2}-\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{4}}{24 \operatorname {Infinity}^{7}}+\frac {\left (24 \operatorname {Infinity}^{5}+154 \operatorname {Infinity}^{4}+102 \operatorname {Infinity}^{3}+19 \operatorname {Infinity}^{2}+\operatorname {Infinity} \right ) \left (x -\operatorname {Infinity} \right )^{5}}{120 \operatorname {Infinity}^{9}}\right ) y^{\prime }\left (\operatorname {Infinity} \right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 124
AsymptoticDSolveValue[x^3*y''[x]+(x^2+x)*y'[x]-y[x]==0,y[x],{x,Infinity,5}]
\[ y(x)\to c_1 \left (\frac {1}{120 x^5}+\frac {1}{24 x^4}+\frac {1}{6 x^3}+\frac {1}{2 x^2}+\frac {1}{x}+1\right )+c_2 \left (-\frac {137}{7200 x^5}-\frac {\log (x)}{120 x^5}-\frac {25}{288 x^4}-\frac {\log (x)}{24 x^4}-\frac {11}{36 x^3}-\frac {\log (x)}{6 x^3}-\frac {3}{4 x^2}-\frac {\log (x)}{2 x^2}-\frac {1}{x}-\frac {\log (x)}{x}-\log (x)\right ) \]