Internal problem ID [8541]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 961.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y^{2}+2 y x +x^{2}+{\mathrm e}^{2+2 y^{4}-4 y^{2} x^{2}+2 x^{4}+2 y^{6}-6 y^{4} x^{2}+6 x^{4} y^{2}-2 x^{6}}}{y^{2}+2 y x +x^{2}-{\mathrm e}^{2+2 y^{4}-4 y^{2} x^{2}+2 x^{4}+2 y^{6}-6 y^{4} x^{2}+6 x^{4} y^{2}-2 x^{6}}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 45
dsolve(diff(y(x),x) = (y(x)^2+2*x*y(x)+x^2+exp(2+2*y(x)^4-4*x^2*y(x)^2+2*x^4+2*y(x)^6-6*x^2*y(x)^4+6*x^4*y(x)^2-2*x^6))/(y(x)^2+2*x*y(x)+x^2-exp(2+2*y(x)^4-4*x^2*y(x)^2+2*x^4+2*y(x)^6-6*x^2*y(x)^4+6*x^4*y(x)^2-2*x^6)),y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\RootOf \left (-\textit {\_Z} +\int _{}^{{\mathrm e}^{2 \textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} x}\frac {1}{{\mathrm e}^{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}+2}+\textit {\_a}}d \textit {\_a} +c_{1}\right )}-x \]
✓ Solution by Mathematica
Time used: 37.859 (sec). Leaf size: 813
DSolve[y'[x] == (E^(2 + 2*x^4 - 2*x^6 - 4*x^2*y[x]^2 + 6*x^4*y[x]^2 + 2*y[x]^4 - 6*x^2*y[x]^4 + 2*y[x]^6) + x^2 + 2*x*y[x] + y[x]^2)/(-E^(2 + 2*x^4 - 2*x^6 - 4*x^2*y[x]^2 + 6*x^4*y[x]^2 + 2*y[x]^4 - 6*x^2*y[x]^4 + 2*y[x]^6) + x^2 + 2*x*y[x] + y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\left (\frac {1}{K[1]+y(x)}-\frac {2 e^{2 K[1]^6+6 y(x)^4 K[1]^2+4 y(x)^2 K[1]^2} K[1]}{e^{2 K[1]^6+6 y(x)^4 K[1]^2+4 y(x)^2 K[1]^2} K[1]^2-e^{2 y(x)^6+2 y(x)^4+6 K[1]^4 y(x)^2+2 K[1]^4+2}-e^{2 K[1]^6+6 y(x)^4 K[1]^2+4 y(x)^2 K[1]^2} y(x)^2}\right )dK[1]+\int _1^{y(x)}\left (-\frac {2 e^{2 x^6+6 K[2]^4 x^2+4 K[2]^2 x^2} K[2]}{-e^{2 x^6+6 K[2]^4 x^2+4 K[2]^2 x^2} x^2+e^{2 K[2]^6+2 K[2]^4+6 x^4 K[2]^2+2 x^4+2}+e^{2 x^6+6 K[2]^4 x^2+4 K[2]^2 x^2} K[2]^2}-\int _1^x\left (-\frac {2 e^{2 K[1]^6+6 K[2]^4 K[1]^2+4 K[2]^2 K[1]^2} K[1] \left (24 K[1]^2 K[2]^3+8 K[1]^2 K[2]\right )}{e^{2 K[1]^6+6 K[2]^4 K[1]^2+4 K[2]^2 K[1]^2} K[1]^2-e^{2 K[2]^6+2 K[2]^4+6 K[1]^4 K[2]^2+2 K[1]^4+2}-e^{2 K[1]^6+6 K[2]^4 K[1]^2+4 K[2]^2 K[1]^2} K[2]^2}+\frac {2 e^{2 K[1]^6+6 K[2]^4 K[1]^2+4 K[2]^2 K[1]^2} K[1] \left (e^{2 K[1]^6+6 K[2]^4 K[1]^2+4 K[2]^2 K[1]^2} \left (24 K[1]^2 K[2]^3+8 K[1]^2 K[2]\right ) K[1]^2-2 e^{2 K[1]^6+6 K[2]^4 K[1]^2+4 K[2]^2 K[1]^2} K[2]-e^{2 K[1]^6+6 K[2]^4 K[1]^2+4 K[2]^2 K[1]^2} K[2]^2 \left (24 K[1]^2 K[2]^3+8 K[1]^2 K[2]\right )-e^{2 K[2]^6+2 K[2]^4+6 K[1]^4 K[2]^2+2 K[1]^4+2} \left (12 K[2]^5+8 K[2]^3+12 K[1]^4 K[2]\right )\right )}{\left (e^{2 K[1]^6+6 K[2]^4 K[1]^2+4 K[2]^2 K[1]^2} K[1]^2-e^{2 K[2]^6+2 K[2]^4+6 K[1]^4 K[2]^2+2 K[1]^4+2}-e^{2 K[1]^6+6 K[2]^4 K[1]^2+4 K[2]^2 K[1]^2} K[2]^2\right )^2}-\frac {1}{(K[1]+K[2])^2}\right )dK[1]+\frac {1}{x+K[2]}\right )dK[2]=c_1,y(x)\right ] \]