83.45.4 problem Ex 4 page 52

Internal problem ID [19476]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter IV. Equations of the first order but not of the first degree
Problem number : Ex 4 page 52
Date solved : Monday, March 31, 2025 at 07:19:55 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.018 (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: 0.492 (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: 1.730 (sec). Leaf size: 44

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

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