89.28.16 problem 16

Internal problem ID [24904]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 229
Problem number : 16
Date solved : Thursday, October 02, 2025 at 10:49:21 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 85

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

\begin{align*} y &= -x -\sqrt {x^{2}+2 c_1} \\ y &= -x +\sqrt {x^{2}+2 c_1} \\ y &= \frac {-c_1 x -\sqrt {2 x^{2} c_1^{2}+1}}{c_1} \\ y &= \frac {-c_1 x +\sqrt {2 x^{2} c_1^{2}+1}}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.282 (sec). Leaf size: 172

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

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

✓ Sympy. Time used: 1.481 (sec). Leaf size: 54

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

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