83.7.4 problem 4

Internal problem ID [21952]
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 : 4
Date solved : Thursday, October 02, 2025 at 08:19:20 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 21

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

\[ y = -\frac {5 \operatorname {LambertW}\left (-\frac {3 \,{\mathrm e}^{\frac {8 x}{5}-\frac {3}{5}-\frac {8 c_1}{5}}}{5}\right )}{6}+x -\frac {1}{2} \]

✓ Mathematica. Time used: 2.171 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to -\frac {5}{6} W\left (-e^{\frac {8 x}{5}-1+c_1}\right )+x-\frac {1}{2}\\ y(x)&\to x-\frac {1}{2} \end{align*}

✓ Sympy. Time used: 8.403 (sec). Leaf size: 253

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

\[ \left [ y{\left (x \right )} = x - \frac {5 W\left (\frac {3 \sqrt [5]{C_{1} e^{8 x}}}{5 e^{\frac {3}{5}}}\right )}{6} - \frac {1}{2}, \ y{\left (x \right )} = x - \frac {5 W\left (\frac {3 \sqrt [5]{C_{1} e^{8 x}} \left (-1 + \sqrt {5} + \sqrt {2} i \sqrt {\sqrt {5} + 5}\right )}{20 e^{\frac {3}{5}}}\right )}{6} - \frac {1}{2}, \ y{\left (x \right )} = x - \frac {5 W\left (- \frac {3 \sqrt [5]{C_{1} e^{8 x}} \left (1 + \sqrt {5} - \sqrt {2} i \sqrt {5 - \sqrt {5}}\right )}{20 e^{\frac {3}{5}}}\right )}{6} - \frac {1}{2}, \ y{\left (x \right )} = x - \frac {5 W\left (- \frac {3 \sqrt [5]{C_{1} e^{8 x}} \left (1 + \sqrt {5} + \sqrt {2} i \sqrt {5 - \sqrt {5}}\right )}{20 e^{\frac {3}{5}}}\right )}{6} - \frac {1}{2}, \ y{\left (x \right )} = x - \frac {5 W\left (- \frac {3 \sqrt [5]{C_{1} e^{8 x}} \left (- \sqrt {5} + 1 + \sqrt {2} i \sqrt {\sqrt {5} + 5}\right )}{20 e^{\frac {3}{5}}}\right )}{6} - \frac {1}{2}\right ] \]