67.25.27 problem 35.4 (m)

Internal problem ID [17022]
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 : 35.4 (m)
Date solved : Thursday, October 02, 2025 at 01:42:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (-x^{4}+x \right ) y^{\prime }+3 x^{3} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 28

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+(-x^4+x)*diff(y(x),x)+3*x^3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 39

ode=x^2*D[y[x],{x,2}]+(x-x^4)*D[y[x],x]+3*x^3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (1-\frac {x^3}{3}\right )+c_2 \left (\frac {x^3}{3}+\left (1-\frac {x^3}{3}\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.397 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**3*y(x) + x**2*Derivative(y(x), (x, 2)) + (-x**4 + x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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