77.18.13 problem 13

Internal problem ID [20508]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (A) at page 53
Problem number : 13
Date solved : Thursday, October 02, 2025 at 06:03:42 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 42

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

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

✓ Sympy. Time used: 0.153 (sec). Leaf size: 26

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

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