Internal problem ID [8371]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 791.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-1+y^{2}-2 y x^{2}+x^{4}-x +x y^{2}-2 y x^{3}+x^{5}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 634
dsolve(diff(y(x),x) = (2*x^2*cosh(1/(x-1))-2*x*cosh(1/(x-1))-1+y(x)^2-2*x^2*y(x)+x^4-x+x*y(x)^2-2*x^3*y(x)+x^5)/(x-1)/cosh(1/(x-1)),y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\frac {\left (\int -\frac {4}{\frac {{\mathrm e}^{\frac {1}{x -1}} x}{x +1}+\frac {{\mathrm e}^{-\frac {1}{x -1}} x}{x +1}-\frac {{\mathrm e}^{\frac {1}{x -1}}}{x +1}-\frac {{\mathrm e}^{-\frac {1}{x -1}}}{x +1}}d x \right ) {\mathrm e}^{\frac {2}{x -1}}}{{\mathrm e}^{\frac {2}{x -1}}+1}} {\mathrm e}^{\frac {4 c_{1} {\mathrm e}^{\frac {2}{x -1}}}{{\mathrm e}^{\frac {2}{x -1}}+1}} {\mathrm e}^{\frac {\int -\frac {4}{\frac {{\mathrm e}^{\frac {1}{x -1}} x}{x +1}+\frac {{\mathrm e}^{-\frac {1}{x -1}} x}{x +1}-\frac {{\mathrm e}^{\frac {1}{x -1}}}{x +1}-\frac {{\mathrm e}^{-\frac {1}{x -1}}}{x +1}}d x}{{\mathrm e}^{\frac {2}{x -1}}+1}} {\mathrm e}^{\frac {4 c_{1}}{{\mathrm e}^{\frac {2}{x -1}}+1}} x^{2}-x^{2}+{\mathrm e}^{\frac {\left (\int -\frac {4}{\frac {{\mathrm e}^{\frac {1}{x -1}} x}{x +1}+\frac {{\mathrm e}^{-\frac {1}{x -1}} x}{x +1}-\frac {{\mathrm e}^{\frac {1}{x -1}}}{x +1}-\frac {{\mathrm e}^{-\frac {1}{x -1}}}{x +1}}d x \right ) {\mathrm e}^{\frac {2}{x -1}}}{{\mathrm e}^{\frac {2}{x -1}}+1}} {\mathrm e}^{\frac {4 c_{1} {\mathrm e}^{\frac {2}{x -1}}}{{\mathrm e}^{\frac {2}{x -1}}+1}} {\mathrm e}^{\frac {\int -\frac {4}{\frac {{\mathrm e}^{\frac {1}{x -1}} x}{x +1}+\frac {{\mathrm e}^{-\frac {1}{x -1}} x}{x +1}-\frac {{\mathrm e}^{\frac {1}{x -1}}}{x +1}-\frac {{\mathrm e}^{-\frac {1}{x -1}}}{x +1}}d x}{{\mathrm e}^{\frac {2}{x -1}}+1}} {\mathrm e}^{\frac {4 c_{1}}{{\mathrm e}^{\frac {2}{x -1}}+1}}+1}{-1+{\mathrm e}^{\frac {\left (\int -\frac {4}{\frac {{\mathrm e}^{\frac {1}{x -1}} x}{x +1}+\frac {{\mathrm e}^{-\frac {1}{x -1}} x}{x +1}-\frac {{\mathrm e}^{\frac {1}{x -1}}}{x +1}-\frac {{\mathrm e}^{-\frac {1}{x -1}}}{x +1}}d x \right ) {\mathrm e}^{\frac {2}{x -1}}}{{\mathrm e}^{\frac {2}{x -1}}+1}} {\mathrm e}^{\frac {4 c_{1} {\mathrm e}^{\frac {2}{x -1}}}{{\mathrm e}^{\frac {2}{x -1}}+1}} {\mathrm e}^{\frac {\int -\frac {4}{\frac {{\mathrm e}^{\frac {1}{x -1}} x}{x +1}+\frac {{\mathrm e}^{-\frac {1}{x -1}} x}{x +1}-\frac {{\mathrm e}^{\frac {1}{x -1}}}{x +1}-\frac {{\mathrm e}^{-\frac {1}{x -1}}}{x +1}}d x}{{\mathrm e}^{\frac {2}{x -1}}+1}} {\mathrm e}^{\frac {4 c_{1}}{{\mathrm e}^{\frac {2}{x -1}}+1}}} \]
✓ Solution by Mathematica
Time used: 11.405 (sec). Leaf size: 109
DSolve[y'[x] == (Sech[(-1 + x)^(-1)]*(-1 - x + x^4 + x^5 - 2*x*Cosh[(-1 + x)^(-1)] + 2*x^2*Cosh[(-1 + x)^(-1)] - 2*x^2*y[x] - 2*x^3*y[x] + y[x]^2 + x*y[x]^2))/(-1 + x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\frac {2 (K[5]+1) \text {sech}\left (\frac {1}{K[5]-1}\right )}{K[5]-1}dK[5]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[6]}\frac {2 (K[5]+1) \text {sech}\left (\frac {1}{K[5]-1}\right )}{K[5]-1}dK[5]\right ) (K[6]+1) \text {sech}\left (\frac {1}{K[6]-1}\right )}{K[6]-1}dK[6]+c_1}+x^2+1 \\ y(x)\to x^2+1 \\ \end{align*}