47.1.16 problem 16

Internal problem ID [9736]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES Page 309
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 06:32:32 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.022 (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: 0.074 (sec). Leaf size: 89

ode=(y[x]+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 {2 x^2+e^{2 c_1}}\\ y(x)&\to -x+\sqrt {2 x^2+e^{2 c_1}}\\ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]+1}{K[1] (K[1]+2)}dK[1]=-\log (x)+c_1,y(x)\right ] \end{align*}

✓ Sympy. Time used: 1.515 (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)
 

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