Internal problem ID [8266]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 686.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class C], [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y^{3} x \,{\mathrm e}^{2 x^{2}}}{y \,{\mathrm e}^{x^{2}}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 85
dsolve(diff(y(x),x) = y(x)^3/(y(x)*exp(x^2)+1)*x*exp(2*x^2),y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (-\tan \left (\RootOf \left (-2 x^{2}-\ln \left (\frac {81 \left (\tan ^{2}\left (\textit {\_Z} \right )\right )}{10}+\frac {81}{10}\right )+2 \ln \left (\frac {9 \tan \left (\textit {\_Z} \right )}{2}-\frac {9}{2}\right )+6 c_{1}-2 \textit {\_Z} \right )\right )+1\right ) {\mathrm e}^{-x^{2}}}{\tan \left (\RootOf \left (-2 x^{2}-\ln \left (\frac {81 \left (\tan ^{2}\left (\textit {\_Z} \right )\right )}{10}+\frac {81}{10}\right )+2 \ln \left (\frac {9 \tan \left (\textit {\_Z} \right )}{2}-\frac {9}{2}\right )+6 c_{1}-2 \textit {\_Z} \right )\right )} \]
✓ Solution by Mathematica
Time used: 7.272 (sec). Leaf size: 68
DSolve[y'[x] == (E^(2*x^2)*x*y[x]^3)/(1 + E^x^2*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\log (y(x))-2 y(x)^2 \left (\frac {\log \left (e^{2 x^2} y(x)^2+2 e^{x^2} y(x)+2\right )}{4 y(x)^2}-\frac {\text {ArcTan}\left (e^{x^2} y(x)+1\right )}{2 y(x)^2}\right )=c_1,y(x)\right ] \]