64.7.13 problem 13

Internal problem ID [13316]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, Section 2.4. Special integrating factors and transformations. Exercises page 67
Problem number : 13
Date solved : Monday, March 31, 2025 at 07:51:19 AM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 2 x +3 y+1+\left (4 x +6 y+1\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (-2\right )&=2 \end{align*}

✓ Maple. Time used: 0.213 (sec). Leaf size: 20

ode:=2*x+3*y(x)+1+(4*x+6*y(x)+1)*diff(y(x),x) = 0; 
ic:=y(-2) = 2; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {1}{3}-\frac {2 x}{3}+\frac {\operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {4}{3}+\frac {x}{3}}}{3}\right )}{2} \]

✓ Mathematica. Time used: 0.934 (sec). Leaf size: 30

ode=(2*x+3*y[x]+1)+(4*x+6*y[x]+1)*D[y[x],x]==0; 
ic={y[-2]==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{6} \left (3 W\left (\frac {2}{3} e^{\frac {x+4}{3}}\right )-4 x+2\right ) \]

✓ Sympy. Time used: 3.160 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x + (4*x + 6*y(x) + 1)*Derivative(y(x), x) + 3*y(x) + 1,0) 
ics = {y(-2): 2} 
dsolve(ode,func=y(x),ics=ics)
 

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