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 : Sunday, March 30, 2025 at 11:23:27 AM
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*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 76

Order:=6; 
ode:=2*x^3*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=infinity);
 

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

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

ode=2*x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,Infinity,5}]
 

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

✓ Sympy. Time used: 1.354 (sec). Leaf size: 110

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**3*Derivative(y(x), (x, 2)) + x**2*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=inf,n=6)
 

\[ y{\left (x \right )} = C_{3} \left (- C_{2} + x - \frac {\left (C_{2} - x\right )^{2}}{4 C_{2}} - \frac {\left (C_{2} - x\right )^{3}}{8 C_{2}^{2}} - \frac {5 \left (C_{2} - x\right )^{4}}{64 C_{2}^{3}} + \frac {\left (C_{2} - x\right )^{3}}{12 C_{2}^{3}} + \frac {7 \left (C_{2} - x\right )^{4}}{48 C_{2}^{4}}\right ) + C_{1} \left (1 - \frac {\left (C_{2} - x\right )^{2}}{4 C_{2}^{3}} - \frac {7 \left (C_{2} - x\right )^{3}}{24 C_{2}^{4}} - \frac {59 \left (C_{2} - x\right )^{4}}{192 C_{2}^{5}} + \frac {\left (C_{2} - x\right )^{4}}{96 C_{2}^{6}}\right ) + O\left (x^{6}\right ) \]