60.2.246 problem 822

Internal problem ID [10833]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 822
Date solved : Monday, January 27, 2025 at 10:11:14 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=\frac {\left (4 \,{\mathrm e}^{-x^{2}}-4 x^{2} {\mathrm e}^{-x^{2}}+4 y^{2}-4 x^{2} {\mathrm e}^{-x^{2}} y+x^{4} {\mathrm e}^{-2 x^{2}}\right ) x}{4} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 35

dsolve(diff(y(x),x) = 1/4*(4*exp(-x^2)-4*x^2*exp(-x^2)+4*y(x)^2-4*x^2*exp(-x^2)*y(x)+x^4*exp(-x^2)^2)*x,y(x), singsol=all)
 

\[ y = \frac {-4+x^{2} \left (x^{2}-2 c_{1} \right ) {\mathrm e}^{-x^{2}}}{2 x^{2}-4 c_{1}} \]

✓ Solution by Mathematica

Time used: 0.486 (sec). Leaf size: 50

DSolve[D[y[x],x] == (x*(4/E^x^2 - (4*x^2)/E^x^2 + x^4/E^(2*x^2) - (4*x^2*y[x])/E^x^2 + 4*y[x]^2))/4,y[x],x,IncludeSingularSolutions -> True]
 

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