20.4.10 problem Problem 18

Internal problem ID [3645]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 18
Date solved : Monday, January 27, 2025 at 07:49:47 AM
CAS classification : [[_homogeneous, `class A`]]

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 2.059 (sec). Leaf size: 38

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

\begin{align*} y(x)\to -x \sqrt {\log (\log (x)+2 c_1)} \\ y(x)\to x \sqrt {\log (\log (x)+2 c_1)} \\ \end{align*}