54.2.420 problem 999

Internal problem ID [12294]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 999
Date solved : Wednesday, October 01, 2025 at 01:40:46 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Riccati]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 36

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

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

✓ Mathematica. Time used: 0.212 (sec). Leaf size: 36

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

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

✓ Sympy. Time used: 1.006 (sec). Leaf size: 34

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

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