77.1.48 problem 67 (page 109)

Internal problem ID [17867]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 67 (page 109)
Date solved : Monday, March 31, 2025 at 04:37:48 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 32

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

\begin{align*} y &= \frac {x^{3}}{3}+c_1 \\ y &= \frac {1}{-x +c_1} \\ y &= c_1 \,{\mathrm e}^{\frac {x^{2}}{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.114 (sec). Leaf size: 48

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

\begin{align*} y(x)\to -\frac {1}{x+c_1} \\ y(x)\to c_1 e^{\frac {x^2}{2}} \\ y(x)\to \frac {x^3}{3}+c_1 \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.439 (sec). Leaf size: 27

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

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