60.2.215 problem 791

Internal problem ID [10802]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 791
Date solved : Tuesday, January 28, 2025 at 05:13:56 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} 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 x^{2} y+x^{4}-x +x y^{2}-2 x^{3} y+x^{5}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )} \end{align*}

✓ Solution by Maple

Time used: 0.025 (sec). Leaf size: 161

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)
 

\[ y = \frac {\left (x^{2} {\mathrm e}^{-4 \left (\int \frac {{\mathrm e}^{\frac {1}{x -1}} \left (x +1\right )}{\left (x -1\right ) \left ({\mathrm e}^{\frac {2}{x -1}}+1\right )}d x \right )+4 c_{1}}-x^{2}+{\mathrm e}^{-4 \left (\int \frac {{\mathrm e}^{\frac {1}{x -1}} \left (x +1\right )}{\left (x -1\right ) \left ({\mathrm e}^{\frac {2}{x -1}}+1\right )}d x \right )+4 c_{1}}+1\right ) {\mathrm e}^{4 \left (\int \frac {{\mathrm e}^{\frac {1}{x -1}} \left (x +1\right )}{\left (x -1\right ) \left ({\mathrm e}^{\frac {2}{x -1}}+1\right )}d x \right )}}{{\mathrm e}^{4 c_{1}}-{\mathrm e}^{4 \left (\int \frac {{\mathrm e}^{\frac {1}{x -1}} \left (x +1\right )}{\left (x -1\right ) \left ({\mathrm e}^{\frac {2}{x -1}}+1\right )}d x \right )}} \]

✓ Solution by Mathematica

Time used: 1.405 (sec). Leaf size: 109

DSolve[D[y[x],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*}