9.23.10 problem 10

Internal problem ID [3360]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 06:37:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 34

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

\[ y = \sqrt {x}\, \left (x \left (1-\frac {1}{16} x^{2}+\frac {7}{2560} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_1 +\left (1-\frac {1}{16} x^{2}+\frac {5}{1536} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 58

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

\[ y(x)\to c_1 \left (\frac {5 x^{9/2}}{1536}-\frac {x^{5/2}}{16}+\sqrt {x}\right )+c_2 \left (\frac {7 x^{11/2}}{2560}-\frac {x^{7/2}}{16}+x^{3/2}\right ) \]

✓ Sympy. Time used: 0.332 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} \left (1 - \frac {x^{2}}{16}\right ) + C_{1} \sqrt {x} \left (\frac {5 x^{4}}{1536} - \frac {x^{2}}{16} + 1\right ) + O\left (x^{6}\right ) \]