56.4.1 problem Ex 1

Internal problem ID [14094]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 11. Equations in which M and N are linear but not homogeneous. Page 16
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:13:06 AM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} \left (x +y+1\right ) y^{\prime }+1+4 x +3 y&=0 \end{align*}

✓ Maple. Time used: 0.076 (sec). Leaf size: 29

ode:=4*x+3*y(x)+1+(x+y(x)+1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -3-\frac {\left (x -2\right ) \left (2 \operatorname {LambertW}\left (c_1 \left (x -2\right )\right )+1\right )}{\operatorname {LambertW}\left (c_1 \left (x -2\right )\right )} \]

✓ Mathematica. Time used: 0.795 (sec). Leaf size: 159

ode=(4*x+3*y[x]+1)+(x+y[x]+1)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {(-2)^{2/3} \left (-2 x \log \left (\frac {3 (-2)^{2/3} (y(x)+2 x-1)}{y(x)+x+1}\right )+(2 x-1) \log \left (-\frac {3 (-2)^{2/3} (x-2)}{y(x)+x+1}\right )+\log \left (\frac {3 (-2)^{2/3} (y(x)+2 x-1)}{y(x)+x+1}\right )+y(x) \left (\log \left (-\frac {3 (-2)^{2/3} (x-2)}{y(x)+x+1}\right )-\log \left (\frac {3 (-2)^{2/3} (y(x)+2 x-1)}{y(x)+x+1}\right )+1\right )+x+1\right )}{9 (y(x)+2 x-1)}=\frac {1}{9} (-2)^{2/3} \log (x-2)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.764 (sec). Leaf size: 19

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

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