44.20.5 problem 3(c)

Internal problem ID [9419]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.5. More on Regular Singular Points. Page 183
Problem number : 3(c)
Date solved : Tuesday, September 30, 2025 at 06:18:32 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x y^{\prime \prime }-y^{\prime }+4 x^{3} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 28

Order:=8; 
ode:=x*diff(diff(y(x),x),x)-diff(y(x),x)+4*x^3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1-\frac {1}{6} x^{4}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (-2+x^{4}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 30

ode=x*D[y[x],{x,2}]-D[y[x],x]+4*x^3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (1-\frac {x^4}{2}\right )+c_2 \left (x^2-\frac {x^6}{6}\right ) \]

✓ Sympy. Time used: 0.235 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**3*y(x) + x*Derivative(y(x), (x, 2)) - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {x^{4}}{2}\right ) + C_{1} x^{2} \left (1 - \frac {x^{4}}{6}\right ) + O\left (x^{8}\right ) \]