82.17.5 problem Ex. 5

Internal problem ID [18750]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Problems at page 37
Problem number : Ex. 5
Date solved : Monday, March 31, 2025 at 06:07:39 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.835 (sec). Leaf size: 57

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

\begin{align*} y(x)\to -\sqrt {1+e^{2 c_1} \left (x^2+1\right )} \\ y(x)\to \sqrt {1+e^{2 c_1} \left (x^2+1\right )} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.616 (sec). Leaf size: 29

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

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