83.37.12 problem 12

Internal problem ID [19386]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (B) at page 128
Problem number : 12
Date solved : Monday, March 31, 2025 at 07:12:11 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: 0.004 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)-1/x^(1/2)*diff(y(x),x)+1/4*y(x)/x^2*(-8+x^(1/2)+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: 0.038 (sec). Leaf size: 30

ode=D[y[x],{x,2}]-1/Sqrt[x]*D[y[x],x]+y[x]/(4*x^2)*(-8+Sqrt[x]+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