60.2.225 problem 801

Internal problem ID [10812]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 801
Date solved : Monday, January 27, 2025 at 10:04:07 PM
CAS classification : [_Abel]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 45

dsolve(diff(y(x),x) = 1/2*(y(x)*exp(-1/4*x^2)*x+2+2*y(x)^2*exp(-1/2*x^2)+2*y(x)^3*exp(-3/4*x^2))*exp(1/4*x^2),y(x), singsol=all)
 

\[ y = \frac {29 \,{\mathrm e}^{\frac {x^{2}}{4}} \operatorname {RootOf}\left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+x +3 c_{1} \right )}{9}-\frac {{\mathrm e}^{\frac {x^{2}}{4}}}{3} \]

✓ Solution by Mathematica

Time used: 0.380 (sec). Leaf size: 104

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

\[ \text {Solve}\left [\int _1^{\frac {3 e^{-\frac {x^2}{2}} y(x)+e^{-\frac {x^2}{4}}}{\sqrt [3]{29} \sqrt [3]{e^{-\frac {3 x^2}{4}}}}}\frac {1}{K[1]^3-\frac {3 K[1]}{29^{2/3}}+1}dK[1]=\frac {1}{9} 29^{2/3} e^{\frac {x^2}{2}} \left (e^{-\frac {3 x^2}{4}}\right )^{2/3} x+c_1,y(x)\right ] \]