60.2.405 problem 982

Internal problem ID [10992]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 982
Date solved : Monday, January 27, 2025 at 10:38:58 PM
CAS classification : [_Abel]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 76

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

\[ -\frac {\ln \left (7\right )}{3}+\frac {\ln \left (1+{\mathrm e}^{-\frac {x^{2}}{2}} y^{2}+y \,{\mathrm e}^{-\frac {x^{2}}{4}}\right )}{3}+\frac {2 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, y \,{\mathrm e}^{-\frac {x^{2}}{4}}}{3}+\frac {\sqrt {3}}{3}\right )}{9}-\frac {2 \ln \left (y \,{\mathrm e}^{-\frac {x^{2}}{4}}\right )}{3}+\frac {2 x}{3}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.392 (sec). Leaf size: 113

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

\[ \text {Solve}\left [\int _1^{\frac {3 e^{-\frac {x^2}{2}} y(x)+e^{-\frac {x^2}{4}}}{\sqrt [3]{7} \sqrt [3]{-e^{-\frac {3 x^2}{4}}}}}\frac {1}{K[1]^3-\frac {6 \sqrt [3]{-1} K[1]}{7^{2/3}}+1}dK[1]=\frac {1}{9} 7^{2/3} e^{\frac {x^2}{2}} \left (-e^{-\frac {3 x^2}{4}}\right )^{2/3} x+c_1,y(x)\right ] \]