29.33.26 problem 989

Internal problem ID [5565]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 989
Date solved : Tuesday, March 04, 2025 at 10:19:18 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.072 (sec). Leaf size: 85

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

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

✓ Mathematica. Time used: 0.604 (sec). Leaf size: 172

ode=(x+y[x])^2 (D[y[x],x])^2 -(x^2-x y[x]-2 y[x]^2) D[y[x],x]-(x-y[x])y[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: 2.294 (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)
 

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