83.18.13 problem 13

Internal problem ID [19144]
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, March 13, 2025 at 01:45:13 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.067 (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: 0.268 (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)
 

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