60.2.359 problem 937

Internal problem ID [10933]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 937
Date solved : Friday, March 14, 2025 at 02:52:13 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {-2 y-2 \ln \left (2 x +1\right )-2+2 x y^{3}+y^{3}+6 y^{2} \ln \left (2 x +1\right ) x +3 y^{2} \ln \left (2 x +1\right )+6 y \ln \left (2 x +1\right )^{2} x +3 y \ln \left (2 x +1\right )^{2}+2 \ln \left (2 x +1\right )^{3} x +\ln \left (2 x +1\right )^{3}}{\left (2 x +1\right ) \left (y+\ln \left (2 x +1\right )+1\right )} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 79

ode:=diff(y(x),x) = 1/(2*x+1)*(-2*y(x)-2*ln(2*x+1)-2+2*x*y(x)^3+y(x)^3+6*y(x)^2*ln(2*x+1)*x+3*y(x)^2*ln(2*x+1)+6*y(x)*ln(2*x+1)^2*x+3*y(x)*ln(2*x+1)^2+2*ln(2*x+1)^3*x+ln(2*x+1)^3)/(y(x)+ln(2*x+1)+1); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-\sqrt {c_{1} -2 x}\, \ln \left (2 x +1\right )+\ln \left (2 x +1\right )+1}{\sqrt {c_{1} -2 x}-1} \\ y &= \frac {-\sqrt {c_{1} -2 x}\, \ln \left (2 x +1\right )-\ln \left (2 x +1\right )-1}{\sqrt {c_{1} -2 x}+1} \\ \end{align*}

✓ Mathematica. Time used: 0.563 (sec). Leaf size: 69

ode=D[y[x],x] == (-2 - 2*Log[1 + 2*x] + Log[1 + 2*x]^3 + 2*x*Log[1 + 2*x]^3 - 2*y[x] + 3*Log[1 + 2*x]^2*y[x] + 6*x*Log[1 + 2*x]^2*y[x] + 3*Log[1 + 2*x]*y[x]^2 + 6*x*Log[1 + 2*x]*y[x]^2 + y[x]^3 + 2*x*y[x]^3)/((1 + 2*x)*(1 + Log[1 + 2*x] + y[x])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 5.907 (sec). Leaf size: 80

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*x*y(x)**3 + 6*x*y(x)**2*log(2*x + 1) + 6*x*y(x)*log(2*x + 1)**2 + 2*x*log(2*x + 1)**3 + y(x)**3 + 3*y(x)**2*log(2*x + 1) + 3*y(x)*log(2*x + 1)**2 - 2*y(x) + log(2*x + 1)**3 - 2*log(2*x + 1) - 2)/((2*x + 1)*(y(x) + log(2*x + 1) + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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