60.2.36 problem 612

Internal problem ID [10623]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 612
Date solved : Monday, January 27, 2025 at 09:18:23 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 38

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

\begin{align*} y &= \operatorname {RootOf}\left (F \left (\textit {\_Z} \,{\mathrm e}^{-\frac {x^{2}}{4}}\right )\right ) \\ y &= \operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) {\mathrm e}^{\frac {x^{2}}{4}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.310 (sec). Leaf size: 199

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

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