81.15.1 problem 19-1

Internal problem ID [21709]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 19. Change of variables. Page 483
Problem number : 19-1
Date solved : Thursday, October 02, 2025 at 08:00:12 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime }&=\frac {x +y+1}{x +2 y+3} \end{align*}

✓ Maple. Time used: 0.496 (sec). Leaf size: 76

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

\[ y = \frac {-2 c_1 -\operatorname {RootOf}\left (-c_1^{2} x^{2}+2 c_1^{2} x -{\mathrm e}^{2 \operatorname {RootOf}\left (c_1^{2} x^{2}+{\mathrm e}^{2 \textit {\_Z}} \cosh \left (\textit {\_Z} \sqrt {2}\right )^{2}-2 c_1^{2} x +c_1^{2}\right )}-c_1^{2}+2 \textit {\_Z}^{2}\right )}{c_1} \]

✓ Mathematica. Time used: 0.064 (sec). Leaf size: 70

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

\[ \text {Solve}\left [2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} (y(x)+x+1)}{2 y(x)+x+3}\right )+c_1=2 \log \left (\frac {2 \left (x^2-2 y(x)^2-8 y(x)-2 x-7\right )}{(x-1)^2}\right )+4 \log (x-1),y(x)\right ] \]

✓ Sympy. Time used: 20.941 (sec). Leaf size: 56

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

\[ \log {\left (y{\left (x \right )} + 2 \right )} = C_{1} - \log {\left (\left (\frac {x - 1}{y{\left (x \right )} + 2} + \sqrt {2}\right )^{\frac {1}{2} - \frac {\sqrt {2}}{4}} \sqrt [4]{\left (\frac {x - 1}{y{\left (x \right )} + 2} - \sqrt {2}\right )^{\sqrt {2} + 2}} \right )} \]