Internal problem ID [8217]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 637.
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 {{\mathrm e}^{-x^{2}} x}{y \,{\mathrm e}^{x^{2}}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 84
dsolve(diff(y(x),x) = 1/(y(x)*exp(x^2)+1)*exp(-x^2)*x,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\tan \left (\RootOf \left (2 x^{2}+2 \ln \left (\frac {9 \tan \left (\textit {\_Z} \right )}{2}-\frac {9}{2}\right )-\ln \left (\frac {81 \left (\tan ^{2}\left (\textit {\_Z} \right )\right )}{10}+\frac {81}{10}\right )+6 c_{1}-2 \textit {\_Z} \right )\right ) {\mathrm e}^{-x^{2}}}{\tan \left (\RootOf \left (2 x^{2}+2 \ln \left (\frac {9 \tan \left (\textit {\_Z} \right )}{2}-\frac {9}{2}\right )-\ln \left (\frac {81 \left (\tan ^{2}\left (\textit {\_Z} \right )\right )}{10}+\frac {81}{10}\right )+6 c_{1}-2 \textit {\_Z} \right )\right )-1} \]
✓ Solution by Mathematica
Time used: 6.991 (sec). Leaf size: 62
DSolve[y'[x] == x/(E^x^2*(1 + E^x^2*y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {1}{2} \text {ArcTan}\left (2 e^{x^2} y(x)+1\right )-\frac {1}{4} \log \left (2 e^{2 x^2} y(x)^2+2 e^{x^2} y(x)+1\right )+\frac {1}{2} \log \left (e^{x^2}\right )=c_1,y(x)\right ] \]