54.2.1 problem 1

Internal problem ID [8533]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number : 1
Date solved : Sunday, March 30, 2025 at 01:13:43 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+4&=0 \end{align*}

✓ Maple. Time used: 0.337 (sec). Leaf size: 53

ode:=x^3*diff(y(x),x)^2+x^2*y(x)*diff(y(x),x)+4 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {4}{\sqrt {x}} \\ y &= \frac {4}{\sqrt {x}} \\ y &= \frac {c_1^{2} x +16}{2 c_1 x} \\ y &= \frac {c_1^{2}+16 x}{2 c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 1.76 (sec). Leaf size: 77

ode=x^3*(D[y[x],x])^2+x^2*y[x]*D[y[x],x]+4==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {e^{-\frac {c_1}{2}} \left (x+16 e^{c_1}\right )}{2 x} \\ y(x)\to \frac {e^{-\frac {c_1}{2}} \left (x+16 e^{c_1}\right )}{2 x} \\ y(x)\to -\frac {4}{\sqrt {x}} \\ y(x)\to \frac {4}{\sqrt {x}} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2*y(x) + sqrt(x**3*(x*y(x)**2 - 16)))/(2*x**3) cannot be solved by the factorable group method