76.13.3 problem Ex. 3

Internal problem ID [20088]
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 32
Problem number : Ex. 3
Date solved : Thursday, October 02, 2025 at 05:22:35 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 65

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

\begin{align*} y &= -x +c_1 \\ y &= \frac {-c_1 x -\sqrt {-3 x^{2} c_1^{2}+4}}{2 c_1} \\ y &= \frac {-c_1 x +\sqrt {-3 x^{2} c_1^{2}+4}}{2 c_1} \\ y &= c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.254 (sec). Leaf size: 130

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

\begin{align*} y(x)&\to \frac {1}{2} \left (-x-\sqrt {-3 x^2+4 e^{c_1}}\right )\\ y(x)&\to \frac {1}{2} \left (-x+\sqrt {-3 x^2+4 e^{c_1}}\right )\\ y(x)&\to c_1\\ y(x)&\to -x+c_1\\ y(x)&\to \frac {1}{2} \left (-\sqrt {3} \sqrt {-x^2}-x\right )\\ y(x)&\to \frac {1}{2} \left (\sqrt {3} \sqrt {-x^2}-x\right ) \end{align*}

✓ Sympy. Time used: 1.155 (sec). Leaf size: 44

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

\[ \left [ y{\left (x \right )} = C_{1} - x, \ y{\left (x \right )} = - \frac {x}{2} - \frac {\sqrt {C_{1} - 3 x^{2}}}{2}, \ y{\left (x \right )} = - \frac {x}{2} + \frac {\sqrt {C_{1} - 3 x^{2}}}{2}, \ y{\left (x \right )} = C_{1}\right ] \]