58.7.3 problem 3

Internal problem ID [14658]
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 : 3
Date solved : Thursday, October 02, 2025 at 09:48:54 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 56

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

\begin{align*} y &= \frac {-1-{\mathrm e}^{-x} \sqrt {{\mathrm e}^{x} \left (-4 c_1 x +{\mathrm e}^{x}\right )}}{2 x} \\ y &= \frac {{\mathrm e}^{-x} \sqrt {{\mathrm e}^{x} \left (-4 c_1 x +{\mathrm e}^{x}\right )}-1}{2 x} \\ \end{align*}

✓ Mathematica. Time used: 2.317 (sec). Leaf size: 69

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

\begin{align*} y(x)&\to -\frac {1+\frac {\sqrt {e^x+4 c_1 x}}{\sqrt {e^x}}}{2 x}\\ y(x)&\to \frac {-1+\frac {\sqrt {e^x+4 c_1 x}}{\sqrt {e^x}}}{2 x} \end{align*}

✓ Sympy. Time used: 2.260 (sec). Leaf size: 87

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

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