89.10.15 problem 15

Internal problem ID [24508]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 77
Problem number : 15
Date solved : Thursday, October 02, 2025 at 10:43:54 PM
CAS classification : [[_homogeneous, `class D`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 72

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

\begin{align*} y &= \frac {-{\mathrm e}^{-3 x} \sqrt {{\mathrm e}^{3 x} c_1 \left ({\mathrm e}^{3 x} c_1 \,x^{2}+1\right )}+c_1 x}{c_1} \\ y &= \frac {{\mathrm e}^{-3 x} \sqrt {{\mathrm e}^{3 x} c_1 \left ({\mathrm e}^{3 x} c_1 \,x^{2}+1\right )}+c_1 x}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 32.327 (sec). Leaf size: 129

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

\begin{align*} y(x)&\to x-e^{-3 x} \sqrt {e^{6 x} x^2-e^{3 x+2 c_1}}\\ y(x)&\to x+e^{-3 x} \sqrt {e^{6 x} x^2-e^{3 x+2 c_1}}\\ y(x)&\to x-e^{-3 x} \sqrt {e^{6 x} x^2}\\ y(x)&\to e^{-3 x} \sqrt {e^{6 x} x^2}+x \end{align*}

✓ Sympy. Time used: 3.372 (sec). Leaf size: 82

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

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