40.17.10 problem 20

Internal problem ID [6811]
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
Date solved : Monday, January 27, 2025 at 02:30:58 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 2 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: 0.005 (sec). Leaf size: 76

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);
 

\[ y = \frac {c_1 \left (1-\frac {1}{x}+\frac {1}{6 x^{2}}-\frac {1}{90 x^{3}}+\frac {1}{2520 x^{4}}-\frac {1}{113400 x^{5}}+O\left (\frac {1}{x^{6}}\right )\right )}{\sqrt {\frac {1}{x}}}+c_2 \left (1-\frac {1}{3 x}+\frac {1}{30 x^{2}}-\frac {1}{630 x^{3}}+\frac {1}{22680 x^{4}}-\frac {1}{1247400 x^{5}}+O\left (\frac {1}{x^{6}}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 96

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

\[ y(x)\to c_2 \left (\frac {1}{6 x^{3/2}}-\frac {1}{90 x^{5/2}}+\frac {1}{2520 x^{7/2}}-\frac {1}{113400 x^{9/2}}+\sqrt {x}-\frac {1}{\sqrt {x}}\right )+c_1 \left (-\frac {1}{1247400 x^5}+\frac {1}{22680 x^4}-\frac {1}{630 x^3}+\frac {1}{30 x^2}-\frac {1}{3 x}+1\right ) \]