88.26.1 problem 1

Internal problem ID [24221]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 7. Series Methods. Exercises at page 220
Problem number : 1
Date solved : Thursday, October 02, 2025 at 10:00:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 32

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{4} x^{2}+\frac {9}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {3}{4} x^{2}+\frac {53}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 60

ode=x^2*D[y[x],{x,2}]+(x-4*x^3)*D[y[x],{x,1}]-x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {9 x^4}{64}+\frac {x^2}{4}+1\right )+c_2 \left (\frac {53 x^4}{128}+\frac {3 x^2}{4}+\left (\frac {9 x^4}{64}+\frac {x^2}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.391 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = C_{1} \left (\frac {9 x^{4}}{64} + \frac {x^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]