48.6.15 problem 15

Internal problem ID [9933]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 06:44:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 32

Order:=8; 
ode:=x*diff(diff(y(x),x),x)+(x^3-1)*diff(y(x),x)+x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1-\frac {1}{5} x^{3}+\frac {1}{40} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (-2+\frac {2}{3} x^{3}-\frac {1}{9} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 44

ode=x*D[y[x],{x,2}]+(x^3-1)*D[y[x],x]+x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^6}{18}-\frac {x^3}{3}+1\right )+c_2 \left (\frac {x^8}{40}-\frac {x^5}{5}+x^2\right ) \]

✓ Sympy. Time used: 0.304 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x) + x*Derivative(y(x), (x, 2)) + (x**3 - 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6}}{18} - \frac {x^{3}}{3} + 1\right ) + C_{1} x^{2} \left (1 - \frac {x^{3}}{5}\right ) + O\left (x^{8}\right ) \]