58.2.28 problem 28

Internal problem ID [9151]
Book : Second order enumerated odes
Section : section 2
Problem number : 28
Date solved : Sunday, March 30, 2025 at 02:24:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {y \left (-8+\sqrt {x}+x \right )}{4 x^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)-1/x^(1/2)*diff(y(x),x)+1/4/x^2*(x+x^(1/2)-8)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\sqrt {x}} \left (c_2 \,x^{3}+c_1 \right )}{x} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 30

ode=D[y[x],{x,2}]-1/x^(1/2)*D[y[x],x]+y[x]/(4*x^2)*(-8+x^(1/2)+x)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{\sqrt {x}} \left (c_2 x^3+3 c_1\right )}{3 x} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + (sqrt(x) + x - 8)*y(x)/(4*x**2) - Derivative(y(x), x)/sqrt(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x) - y(x)/(4*x) - y(x)/(4*sqrt(x)) + 2*y(x)/x**(3/2) cannot be solved by the factorable group method