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 : Monday, January 27, 2025 at 09:01:11 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 54

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

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

✓ Solution by Mathematica

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

DSolve[D[y[x],x]==(x^3*Exp[x^2])/(y[x]*Log[y[x]]),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*}