60.2.369 problem 946

Internal problem ID [10956]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 946
Date solved : Tuesday, January 28, 2025 at 05:35:48 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {\left (-8 \,{\mathrm e}^{-x^{2}} y+4 x^{2} {\mathrm e}^{-2 x^{2}}-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}} y-4 x^{4} {\mathrm e}^{-2 x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{-8 y+4 x^{2} {\mathrm e}^{-x^{2}}-8} \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 85

dsolve(diff(y(x),x) = (-8*exp(-x^2)*y(x)+4*x^2*exp(-x^2)^2-8*exp(-x^2)+8*x^2*exp(-x^2)*y(x)-4*x^4*exp(-x^2)^2+8*x^2*exp(-x^2)-8*y(x)^3+12*x^2*exp(-x^2)*y(x)^2-6*y(x)*x^4*exp(-x^2)^2+x^6*exp(-x^2)^3)*x/(-8*y(x)+4*x^2*exp(-x^2)-8),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.117 (sec). Leaf size: 93

DSolve[D[y[x],x] == (x*(-8/E^x^2 + (4*x^2)/E^(2*x^2) + (8*x^2)/E^x^2 - (4*x^4)/E^(2*x^2) + x^6/E^(3*x^2) - (8*y[x])/E^x^2 + (8*x^2*y[x])/E^x^2 - (6*x^4*y[x])/E^(2*x^2) + (12*x^2*y[x]^2)/E^x^2 - 8*y[x]^3))/(-8 + (4*x^2)/E^x^2 - 8*y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{2} e^{-x^2} x^2+\frac {8}{-8+\sqrt {-64 x^2+c_1}} \\ y(x)\to \frac {1}{2} e^{-x^2} x^2-\frac {8}{8+\sqrt {-64 x^2+c_1}} \\ y(x)\to \frac {1}{2} e^{-x^2} x^2 \\ \end{align*}