78.8.29 problem 29

Internal problem ID [18227]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 29
Date solved : Tuesday, January 28, 2025 at 11:40:54 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.379 (sec). Leaf size: 34

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

\[ \text {Solve}\left [\log \left (\frac {y(x)}{x}\right )-\log \left (e^{\frac {x^2}{y(x)^2}}+1\right )=-\log (x)+c_1,y(x)\right ] \]