30.14.6 problem 6

Internal problem ID [7655]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 04:55:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}+1\right ) y^{\prime \prime }-x y^{\prime }+2 x^{2} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

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

Order:=6; 
ode:=(x^3+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+2*x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\left (x -1\right )^{3}}{6}+\frac {7 \left (x -1\right )^{4}}{48}+\frac {7 \left (x -1\right )^{5}}{240}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2}}{4}-\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{4}}{8}+\frac {\left (x -1\right )^{5}}{12}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 78

ode=(1+x^3)*D[y[x],{x,2}]-x*D[y[x],x]+2*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_1 \left (-\frac {1}{20} (x-1)^5+\frac {1}{8} (x-1)^4-\frac {1}{2} (x-1)^2+1\right )+c_2 \left (\frac {19}{240} (x-1)^5-\frac {1}{24} (x-1)^4-\frac {1}{6} (x-1)^3+\frac {1}{4} (x-1)^2+x-1\right ) \]

✓ Sympy. Time used: 0.389 (sec). Leaf size: 80

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*y(x) - x*Derivative(y(x), x) + (x**3 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = - \frac {5 \left (x - 1\right )^{4} r{\left (3 \right )}}{8} + \frac {3 \left (x - 1\right )^{5} r{\left (3 \right )}}{40} + C_{2} \left (x + \frac {23 \left (x - 1\right )^{5}}{240} - \frac {11 \left (x - 1\right )^{4}}{48} + \frac {\left (x - 1\right )^{2}}{4} - 1\right ) + C_{1} \left (\frac {\left (x - 1\right )^{5}}{60} + \frac {\left (x - 1\right )^{4}}{6} - \left (x - 1\right )^{2} + 1\right ) + O\left (x^{6}\right ) \]