Internal problem ID [8197]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 617.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {F \left (\frac {\left (3+y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 92
dsolve(diff(y(x),x) = 1/9*F(1/3*(3+y(x))*exp(3/2*x^2)/y(x))*x*y(x)^2*exp(3*x^2)/exp(9/2*x^2),y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (F \left (\frac {\left (3+\textit {\_Z} \right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 \textit {\_Z}}\right ) \textit {\_Z} \,{\mathrm e}^{3 x^{2}}-9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} \textit {\_Z} -27 \,{\mathrm e}^{\frac {9 x^{2}}{2}}\right ) \\ y \relax (x ) = -\frac {3 \,{\mathrm e}^{\frac {3 x^{2}}{2}}}{{\mathrm e}^{\frac {3 x^{2}}{2}}-3 \RootOf \left (-x^{2}-18 \left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )-27 \textit {\_a}}d \textit {\_a} \right )+2 c_{1}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.628 (sec). Leaf size: 615
DSolve[y'[x] == (x*F[(E^((3*x^2)/2)*(3 + y[x]))/(3*y[x])]*y[x]^2)/(9*E^((3*x^2)/2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {9 e^{\frac {3 x^2}{2}}-F\left (\frac {e^{\frac {3 x^2}{2}} (K[2]+3)}{3 K[2]}\right )}{3 \left (\left (9 e^{\frac {3 x^2}{2}}-F\left (\frac {e^{\frac {3 x^2}{2}} (K[2]+3)}{3 K[2]}\right )\right ) K[2]+27 e^{\frac {3 x^2}{2}}\right )}-\int _1^x\left (-\frac {K[2] \left (\frac {e^{\frac {3 K[1]^2}{2}}}{3 K[2]}-\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]^2}\right ) F'\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[1]}{-9 e^{\frac {3 K[1]^2}{2}} K[2]+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2]-27 e^{\frac {3 K[1]^2}{2}}}+\frac {F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2] \left (F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right )-9 e^{\frac {3 K[1]^2}{2}}+K[2] \left (\frac {e^{\frac {3 K[1]^2}{2}}}{3 K[2]}-\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]^2}\right ) F'\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right )\right ) K[1]}{\left (-9 e^{\frac {3 K[1]^2}{2}} K[2]+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2]-27 e^{\frac {3 K[1]^2}{2}}\right )^2}-\frac {F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[1]}{-9 e^{\frac {3 K[1]^2}{2}} K[2]+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2]-27 e^{\frac {3 K[1]^2}{2}}}\right )dK[1]-\frac {1}{3 K[2]}\right )dK[2]+\int _1^x-\frac {F\left (\frac {e^{\frac {3 K[1]^2}{2}} (y(x)+3)}{3 y(x)}\right ) K[1] y(x)}{-9 e^{\frac {3 K[1]^2}{2}} y(x)+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (y(x)+3)}{3 y(x)}\right ) y(x)-27 e^{\frac {3 K[1]^2}{2}}}dK[1]=c_1,y(x)\right ] \]