9.3.9 problem 9

Internal problem ID [2902]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 7, page 28
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 06:01:50 AM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} x +2 y+\left (3 x +6 y+3\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 23

ode:=x+2*y(x)+(3*x+6*y(x)+3)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 2.397 (sec). Leaf size: 43

ode=(x+2*y[x])+(3*x+6*y[x]+3)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (-2 W\left (-e^{-\frac {x}{6}-1+c_1}\right )-x-3\right )\\ y(x)&\to \frac {1}{2} (-x-3) \end{align*}

✓ Sympy. Time used: 6.275 (sec). Leaf size: 202

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x + (3*x + 6*y(x) + 3)*Derivative(y(x), x) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {x}{2} - W\left (- \frac {\sqrt [6]{C_{1} e^{- x}}}{2 e^{\frac {3}{2}}}\right ) - \frac {3}{2}, \ y{\left (x \right )} = - \frac {x}{2} - W\left (\frac {\sqrt [6]{C_{1} e^{- x}}}{2 e^{\frac {3}{2}}}\right ) - \frac {3}{2}, \ y{\left (x \right )} = - \frac {x}{2} - W\left (- \frac {\sqrt [6]{C_{1} e^{- x}} \left (1 - \sqrt {3} i\right )}{4 e^{\frac {3}{2}}}\right ) - \frac {3}{2}, \ y{\left (x \right )} = - \frac {x}{2} - W\left (\frac {\sqrt [6]{C_{1} e^{- x}} \left (1 - \sqrt {3} i\right )}{4 e^{\frac {3}{2}}}\right ) - \frac {3}{2}, \ y{\left (x \right )} = - \frac {x}{2} - W\left (- \frac {\sqrt [6]{C_{1} e^{- x}} \left (1 + \sqrt {3} i\right )}{4 e^{\frac {3}{2}}}\right ) - \frac {3}{2}, \ y{\left (x \right )} = - \frac {x}{2} - W\left (\frac {\sqrt [6]{C_{1} e^{- x}} \left (1 + \sqrt {3} i\right )}{4 e^{\frac {3}{2}}}\right ) - \frac {3}{2}\right ] \]