60.2.293 problem 871

Internal problem ID [10867]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 871
Date solved : Sunday, March 30, 2025 at 07:17:42 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=\frac {2 x y^{2}+4 y \ln \left (2 x +1\right ) x +2 \ln \left (2 x +1\right )^{2} x +y^{2}-2+\ln \left (2 x +1\right )^{2}+2 y \ln \left (2 x +1\right )}{2 x +1} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

ode:=diff(y(x),x) = 1/(2*x+1)*(2*x*y(x)^2+4*y(x)*ln(2*x+1)*x+2*ln(2*x+1)^2*x+y(x)^2-2+ln(2*x+1)^2+2*y(x)*ln(2*x+1)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-1+\left (c_1 -x \right ) \ln \left (2 x +1\right )}{-c_1 +x} \]

✓ Mathematica. Time used: 0.439 (sec). Leaf size: 34

ode=D[y[x],x] == (-2 + Log[1 + 2*x]^2 + 2*x*Log[1 + 2*x]^2 + 2*Log[1 + 2*x]*y[x] + 4*x*Log[1 + 2*x]*y[x] + y[x]^2 + 2*x*y[x]^2)/(1 + 2*x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\log (2 x+1)+\frac {1}{-x+c_1} \\ y(x)\to -\log (2 x+1) \\ \end{align*}

✓ Sympy. Time used: 1.496 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = \frac {- C_{1} \log {\left (2 x + 1 \right )} - x \log {\left (2 x + 1 \right )} - 1}{C_{1} + x} \]