67.25.8 problem 35.3 (b)

Internal problem ID [17003]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.3 (b)
Date solved : Thursday, October 02, 2025 at 01:41:43 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 76

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

\[ y = \left (1-\frac {\left (x -2\right )^{2}}{16}+\frac {\left (x -2\right )^{3}}{24}-\frac {35 \left (x -2\right )^{4}}{1536}+\frac {89 \left (x -2\right )^{5}}{7680}\right ) y \left (2\right )+\left (x -2-\frac {\left (x -2\right )^{2}}{4}+\frac {\left (x -2\right )^{3}}{16}-\frac {\left (x -2\right )^{4}}{96}-\frac {19 \left (x -2\right )^{5}}{7680}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (\frac {89 (x-2)^5}{7680}-\frac {35 (x-2)^4}{1536}+\frac {1}{24} (x-2)^3-\frac {1}{16} (x-2)^2+1\right )+c_2 \left (-\frac {19 (x-2)^5}{7680}-\frac {1}{96} (x-2)^4+\frac {1}{16} (x-2)^3-\frac {1}{4} (x-2)^2+x-2\right ) \]

✓ Sympy. Time used: 0.275 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(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=2,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x - \frac {\left (x - 2\right )^{4}}{96} + \frac {\left (x - 2\right )^{3}}{16} - \frac {\left (x - 2\right )^{2}}{4} - 2\right ) + C_{1} \left (- \frac {35 \left (x - 2\right )^{4}}{1536} + \frac {\left (x - 2\right )^{3}}{24} - \frac {\left (x - 2\right )^{2}}{16} + 1\right ) + O\left (x^{6}\right ) \]