60.2.350 problem 927

Internal problem ID [10937]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 927
Date solved : Monday, January 27, 2025 at 10:25:45 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 45

dsolve(diff(y(x),x) = -1/8*(-8*exp(-x^2)+8*x^2*exp(-x^2)-8-8*y(x)^2+8*x^2*exp(-x^2)*y(x)-2*x^4*exp(-x^2)^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,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-x^{2}} x^{2}}{2}-\frac {1}{3}+\frac {29 \operatorname {RootOf}\left (x^{2}-162 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+6 c_{1} \right )}{9} \]

✓ Solution by Mathematica

Time used: 0.461 (sec). Leaf size: 90

DSolve[D[y[x],x] == -1/8*(x*(-8 - 8/E^x^2 + (8*x^2)/E^x^2 - (2*x^4)/E^(2*x^2) + x^6/E^(3*x^2) + (8*x^2*y[x])/E^x^2 - (6*x^4*y[x])/E^(2*x^2) - 8*y[x]^2 + (12*x^2*y[x]^2)/E^x^2 - 8*y[x]^3)),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {1}{2} e^{-x^2} x \left (2 e^{x^2}-3 x^2\right )+3 x y(x)}{\sqrt [3]{29} \sqrt [3]{x^3}}}\frac {1}{K[1]^3-\frac {3 K[1]}{29^{2/3}}+1}dK[1]=\frac {1}{18} 29^{2/3} \left (x^3\right )^{2/3}+c_1,y(x)\right ] \]