89.6.46 problem 47

Internal problem ID [24429]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 47
Date solved : Thursday, October 02, 2025 at 10:29:20 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

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

With initial conditions

\begin{align*} y \left (3\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 15

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

\[ y = -\frac {1}{4}+\frac {\sqrt {8 x +1}}{4} \]

✓ Mathematica. Time used: 0.325 (sec). Leaf size: 20

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

\begin{align*} y(x)&\to \frac {1}{4} \left (\sqrt {8 x+1}-1\right ) \end{align*}

✓ Sympy. Time used: 7.973 (sec). Leaf size: 82

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

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