76.4.15 problem 19

Internal problem ID [17409]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.6 (Exact equations and integrating factors). Problems at page 100
Problem number : 19
Date solved : Tuesday, January 28, 2025 at 10:06:05 AM
CAS classification : [_separable]

\begin{align*} x^{2} y^{3}+x \left (1+y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 37

dsolve((x^2*y(x)^3) + x*(1+y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 3.836 (sec). Leaf size: 46

DSolve[(x^2*y[x]^3)+x*(1+y[x]^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{\sqrt {W\left (e^{x^2-2 c_1}\right )}} \\ y(x)\to \frac {1}{\sqrt {W\left (e^{x^2-2 c_1}\right )}} \\ y(x)\to 0 \\ \end{align*}