12.6.16 problem 16

Internal problem ID [1695]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Exact equations. Section 2.5 Page 79
Problem number : 16
Date solved : Saturday, March 29, 2025 at 11:33:30 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} {\mathrm e}^{x y} \left (x^{4} y+4 x^{3}\right )+3 y+\left (x^{5} {\mathrm e}^{x y}+3 x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 25

ode:=exp(x*y(x))*(x^4*y(x)+4*x^3)+3*y(x)+(x^5*exp(x*y(x))+3*x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-3 \operatorname {LambertW}\left (\frac {x^{4} {\mathrm e}^{-\frac {c_1}{3}}}{3}\right )-c_1}{3 x} \]

✓ Mathematica. Time used: 4.187 (sec). Leaf size: 33

ode=(Exp[x*y[x]]*(x^4*y[x]+4*x^3)+3*y[x])+(x^5*Exp[x*y[x]]+3*x)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {c_1-3 W\left (\frac {1}{3} e^{\frac {c_1}{3}} x^4\right )}{3 x} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**4*y(x) + 4*x**3)*exp(x*y(x)) + (x**5*exp(x*y(x)) + 3*x)*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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