83.7.3 problem 3

Internal problem ID [21951]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. VIII at page 53
Problem number : 3
Date solved : Thursday, October 02, 2025 at 08:19:09 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 4 x +3 y+2+\left (5 x +4 y+1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 31

ode:=4*x+3*y(x)+2+(5*x+4*y(x)+1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-4 x +4\right ) \operatorname {LambertW}\left (-\frac {c_1 \left (x +5\right )}{4}\right )+x +5}{4 \operatorname {LambertW}\left (-\frac {c_1 \left (x +5\right )}{4}\right )} \]

✓ Mathematica. Time used: 0.507 (sec). Leaf size: 175

ode=(4*x+3*y[x]+2)+(5*x+4*y[x]+1)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {-4 x \log \left (\frac {12\ 2^{2/3} (y(x)+x-1)}{4 y(x)+5 x+1}\right )+4 (x-1) \log \left (\frac {3\ 2^{2/3} (x+5)}{4 y(x)+5 x+1}\right )+4 \log \left (\frac {12\ 2^{2/3} (y(x)+x-1)}{4 y(x)+5 x+1}\right )+4 y(x) \left (\log \left (\frac {3\ 2^{2/3} (x+5)}{4 y(x)+5 x+1}\right )-\log \left (\frac {12\ 2^{2/3} (y(x)+x-1)}{4 y(x)+5 x+1}\right )+1\right )+5 x+1}{18 \sqrt [3]{2} (y(x)+x-1)}=\frac {1}{9} 2^{2/3} \log (x+5)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.899 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x + (5*x + 4*y(x) + 1)*Derivative(y(x), x) + 3*y(x) + 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - x + e^{C_{1} + W\left (\frac {\left (x + 5\right ) e^{- C_{1}}}{4}\right )} + 1 \]