54.2.214 problem 791

Internal problem ID [12088]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 791
Date solved : Wednesday, October 01, 2025 at 12:24:31 AM
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*}

✓ Maple. Time used: 0.120 (sec). Leaf size: 92

ode:=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)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-x^{2} {\mathrm e}^{2 \int \frac {\left (x +1\right ) \operatorname {sech}\left (\frac {1}{x -1}\right )}{x -1}d x}+x^{2} {\mathrm e}^{4 c_1}+{\mathrm e}^{2 \int \frac {\left (x +1\right ) \operatorname {sech}\left (\frac {1}{x -1}\right )}{x -1}d x}+{\mathrm e}^{4 c_1}}{-{\mathrm e}^{2 \int \frac {\left (x +1\right ) \operatorname {sech}\left (\frac {1}{x -1}\right )}{x -1}d x}+{\mathrm e}^{4 c_1}} \]

✓ Mathematica. Time used: 0.877 (sec). Leaf size: 109

ode=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); 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**5 + x**4 - 2*x**3*y(x) - 2*x**2*y(x) + 2*x**2*cosh(1/(x - 1)) + x*y(x)**2 - 2*x*cosh(1/(x - 1)) - x + y(x)**2 - 1)/((x - 1)*cosh(1/(x - 1))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

IndexError : Index out of range: a[1]