54.2.101 problem 677

Internal problem ID [11975]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 677
Date solved : Tuesday, September 30, 2025 at 11:51:04 PM
CAS classification : [[_homogeneous, `class D`], _Riccati]

\begin{align*} y^{\prime }&=\frac {y+x^{3} a \ln \left (x +1\right )+a \,x^{4}+x^{3} a -x y^{2} \ln \left (x +1\right )-x^{2} y^{2}-x y^{2}}{x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 48

ode:=diff(y(x),x) = (y(x)+x^3*a*ln(1+x)+a*x^4+a*x^3-x*y(x)^2*ln(1+x)-x^2*y(x)^2-x*y(x)^2)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \tanh \left (\frac {\sqrt {a}\, \left (6 \ln \left (x +1\right ) x^{2}+4 x^{3}+3 x^{2}-6 \ln \left (x +1\right )+12 c_1 +6 x +9\right )}{12}\right ) x \sqrt {a} \]

✓ Mathematica. Time used: 0.105 (sec). Leaf size: 57

ode=D[y[x],x] == (a*x^3 + a*x^4 + a*x^3*Log[1 + x] + y[x] - x*y[x]^2 - x^2*y[x]^2 - x*Log[1 + x]*y[x]^2)/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{a-K[1]^2}dK[1]=\frac {1}{12} \left (x \left (4 x^2+3 x+6\right )+6 \left (x^2-1\right ) \log (x+1)\right )+c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE -a*x**3 - a*x**2*log(x + 1) - a*x**2 + x*y(x)**2 + y(x)**2*log(x + 1) + y(x)**2 + Derivative(y(x), x) - y(x)/x cannot be solved by the factorable group method