12.5.52 problem 51

Internal problem ID [1676]
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 : 51
Date solved : Monday, January 27, 2025 at 05:24:59 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`], [_Abel, `2nd type`, `class A`]]

\begin{align*} \left (y+{\mathrm e}^{x^{2}}\right ) y^{\prime }&=2 x \left (y^{2}+y \,{\mathrm e}^{x^{2}}+{\mathrm e}^{2 x^{2}}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 47

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

\begin{align*} y &= \left (-1-\sqrt {2 x^{2}-2 c_1 +1}\right ) {\mathrm e}^{x^{2}} \\ y &= \left (-1+\sqrt {2 x^{2}-2 c_1 +1}\right ) {\mathrm e}^{x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.723 (sec). Leaf size: 76

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

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