54.2.396 problem 975

Internal problem ID [12270]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 975
Date solved : Wednesday, October 01, 2025 at 01:26:34 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Abel]

\begin{align*} y^{\prime }&=y^{3}+x^{2} y^{2}+\frac {y x^{4}}{3}+\frac {x^{6}}{27}-\frac {2 x}{3} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 59

ode:=diff(y(x),x) = y(x)^3+x^2*y(x)^2+1/3*y(x)*x^4+1/27*x^6-2/3*x; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {x^{2} \sqrt {-54 c_1 -2 x}-3}{3 \sqrt {-54 c_1 -2 x}} \\ y &= -\frac {x^{2} \sqrt {-54 c_1 -2 x}+3}{3 \sqrt {-54 c_1 -2 x}} \\ \end{align*}

✓ Mathematica. Time used: 0.121 (sec). Leaf size: 58

ode=D[y[x],x] == (-2*x)/3 + x^6/27 + (x^4*y[x])/3 + x^2*y[x]^2 + y[x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.779 (sec). Leaf size: 66

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**6/27 - x**4*y(x)/3 - x**2*y(x)**2 + 2*x/3 - y(x)**3 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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