12.5.24 problem 21

Internal problem ID [1648]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 21
Date solved : Monday, January 27, 2025 at 05:16:08 AM
CAS classification : [[_homogeneous, `class A`]]

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 26

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

\begin{align*} y &= \sqrt {\ln \left (\ln \left (x \right )+c_1 \right )}\, x \\ y &= -\sqrt {\ln \left (\ln \left (x \right )+c_1 \right )}\, x \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.010 (sec). Leaf size: 38

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