40.17.11 problem 21

Internal problem ID [6812]
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
Date solved : Monday, January 27, 2025 at 02:30:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 112

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);
 

\[ y = \frac {7200 \left (O\left (\frac {1}{x^{6}}\right ) x^{5}+x^{5}+x^{4}+\frac {x^{3}}{2}+\frac {x^{2}}{6}+\frac {x}{24}+\frac {1}{120}\right ) c_2 \ln \left (\frac {1}{x}\right )+7200 x^{5} \left (c_1 +c_2 \right ) O\left (\frac {1}{x^{6}}\right )+7200 c_1 \,x^{5}+\left (7200 c_1 -7200 c_2 \right ) x^{4}+\left (3600 c_1 -5400 c_2 \right ) x^{3}+\left (1200 c_1 -2200 c_2 \right ) x^{2}+\left (300 c_1 -625 c_2 \right ) x +60 c_1 -137 c_2}{7200 x^{5}} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 124

AsymptoticDSolveValue[x^3*D[y[x],{x,2}]+(x^2+x)*D[y[x],x]-y[x]==0,y[x],{x,Infinity,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {1}{120 x^5}+\frac {1}{24 x^4}+\frac {1}{6 x^3}+\frac {1}{2 x^2}+\frac {1}{x}+1\right )+c_2 \left (-\frac {137}{7200 x^5}-\frac {\log (x)}{120 x^5}-\frac {25}{288 x^4}-\frac {\log (x)}{24 x^4}-\frac {11}{36 x^3}-\frac {\log (x)}{6 x^3}-\frac {3}{4 x^2}-\frac {\log (x)}{2 x^2}-\frac {1}{x}-\frac {\log (x)}{x}-\log (x)\right ) \]