29.35.19 problem 1052

Internal problem ID [5617]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 35
Problem number : 1052
Date solved : Sunday, March 30, 2025 at 09:20:17 AM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{3}-\left (x^{2}+x y^{2}+y^{4}\right ) {y^{\prime }}^{2}+x y^{2} \left (x^{2}+x y^{2}+y^{4}\right ) y^{\prime }-x^{3} y^{6}&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 73

ode:=diff(y(x),x)^3-(x^2+x*y(x)^2+y(x)^4)*diff(y(x),x)^2+x*y(x)^2*(x^2+x*y(x)^2+y(x)^4)*diff(y(x),x)-x^3*y(x)^6 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{3}}{3}+c_1 \\ y &= \frac {1}{\left (-3 x +c_1 \right )^{{1}/{3}}} \\ y &= -\frac {1+i \sqrt {3}}{2 \left (-3 x +c_1 \right )^{{1}/{3}}} \\ y &= \frac {i \sqrt {3}-1}{2 \left (-3 x +c_1 \right )^{{1}/{3}}} \\ y &= -\frac {2}{x^{2}-2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.251 (sec). Leaf size: 110

ode=(D[y[x],x])^3 -(x^2+x y[x]^2+ y[x]^4) (D[y[x],x])^2 +x y[x]^2(x^2 +x y[x]^2+ y[x]^4) D[y[x],x]-x^3 y[x]^6==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 3.189 (sec). Leaf size: 88

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*y(x)**6 + x*(x**2 + x*y(x)**2 + y(x)**4)*y(x)**2*Derivative(y(x), x) - (x**2 + x*y(x)**2 + y(x)**4)*Derivative(y(x), x)**2 + Derivative(y(x), x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} + \frac {x^{3}}{3}, \ y{\left (x \right )} = \sqrt [3]{- \frac {1}{C_{1} + 3 x}}, \ y{\left (x \right )} = \frac {\sqrt [3]{- \frac {1}{C_{1} + x}} \left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right )}{6}, \ y{\left (x \right )} = \frac {\sqrt [3]{- \frac {1}{C_{1} + x}} \left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right )}{6}, \ y{\left (x \right )} = - \frac {2}{C_{1} + x^{2}}\right ] \]