15.8.39 problem 41

Internal problem ID [3042]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 41
Date solved : Monday, January 27, 2025 at 07:11:16 AM
CAS classification : [_separable]

\begin{align*} x \,{\mathrm e}^{-y^{2}}+y y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 1.981 (sec). Leaf size: 29

dsolve([(x*exp(-y(x)^2))+y(x)*diff(y(x),x)=0,y(0) = 0],y(x), singsol=all)
 

\begin{align*} y &= \sqrt {\ln \left (-x^{2}+1\right )} \\ y &= -\sqrt {\ln \left (-x^{2}+1\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.165 (sec). Leaf size: 35

DSolve[{(x*Exp[-y[x]^2])+y[x]*D[y[x],x]==0,{y[0]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {\log \left (1-x^2\right )} \\ y(x)\to \sqrt {\log \left (1-x^2\right )} \\ \end{align*}