28.1.2 problem 2

Internal problem ID [4308]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 2
Date solved : Sunday, March 30, 2025 at 02:58:21 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {x^{3} {\mathrm e}^{x^{2}}}{y \ln \left (y\right )} \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 54

ode:=diff(y(x),x) = x^3*exp(x^2)/y(x)/ln(y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {2}\, \sqrt {\frac {x^{2} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}}+2 c_1}{\operatorname {LambertW}\left (2 \left (x^{2} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}}+2 c_1 \right ) {\mathrm e}^{-1}\right )}} \]

✓ Mathematica. Time used: 60.195 (sec). Leaf size: 106

ode=D[y[x],x]==(x^3*Exp[x^2])/(y[x]*Log[y[x]]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt {2 e^{x^2} \left (x^2-1\right )+4 c_1}}{\sqrt {W\left (\frac {2 e^{x^2} \left (x^2-1\right )+4 c_1}{e}\right )}} \\ y(x)\to \frac {\sqrt {2 e^{x^2} \left (x^2-1\right )+4 c_1}}{\sqrt {W\left (\frac {2 e^{x^2} \left (x^2-1\right )+4 c_1}{e}\right )}} \\ \end{align*}

✓ Sympy. Time used: 0.847 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = e^{\frac {W\left (C_{1} + 2 x^{2} e^{x^{2} - 1} - 2 e^{x^{2} - 1}\right )}{2} + \frac {1}{2}} \]