52.2.15 problem 16

Internal problem ID [10438]
Book : Second order enumerated odes
Section : section 2
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 07:26:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 50

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) = x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {560 x^{{3}/{2}}+28 x^{{5}/{2}}+\left (c_1 \,x^{3}+c_2 \right ) {\mathrm e}^{\sqrt {x}}+4 x^{3}+140 x^{2}+1680 x +3360 \sqrt {x}+3360}{x} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 60

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

\begin{align*} y(x)&\to \frac {e^{\sqrt {x}} \left (x^3 \int _1^xe^{-\sqrt {K[1]}}dK[1]+c_2 x^3+2 \Gamma \left (8,\sqrt {x}\right )+3 c_1\right )}{3 x} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + 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 x**(3/2) - 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