60.2.317 problem 895

Internal problem ID [10891]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 895
Date solved : Sunday, March 30, 2025 at 07:20:26 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {\left (-256 a \,x^{2} y-32 a^{2} x^{6}-256 a \,x^{2}+512 y^{3}+192 x^{4} a y^{2}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512 y+64 a \,x^{4}+512} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 88

ode:=diff(y(x),x) = (-256*a*x^2*y(x)-32*a^2*x^6-256*a*x^2+512*y(x)^3+192*x^4*a*y(x)^2+24*y(x)*a^2*x^8+a^3*x^12)*x/(512*y(x)+64*a*x^4+512); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {a \,x^{4}}{8} \\ y &= \frac {8+\left (-\sqrt {-x^{2}+c_1}+1\right ) a \,x^{4}}{8 \sqrt {-x^{2}+c_1}-8} \\ y &= \frac {-8+\left (-\sqrt {-x^{2}+c_1}-1\right ) a \,x^{4}}{8 \sqrt {-x^{2}+c_1}+8} \\ \end{align*}

✓ Mathematica. Time used: 0.483 (sec). Leaf size: 75

ode=D[y[x],x] == (x*(-256*a*x^2 - 32*a^2*x^6 + a^3*x^12 - 256*a*x^2*y[x] + 24*a^2*x^8*y[x] + 192*a*x^4*y[x]^2 + 512*y[x]^3))/(512 + 64*a*x^4 + 512*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {a x^4}{8}+\frac {512}{-512+\sqrt {-262144 x^2+c_1}} \\ y(x)\to -\frac {a x^4}{8}-\frac {512}{512+\sqrt {-262144 x^2+c_1}} \\ y(x)\to -\frac {a x^4}{8} \\ \end{align*}

✓ Sympy. Time used: 3.050 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-x*(a**3*x**12 + 24*a**2*x**8*y(x) - 32*a**2*x**6 + 192*a*x**4*y(x)**2 - 256*a*x**2*y(x) - 256*a*x**2 + 512*y(x)**3)/(64*a*x**4 + 512*y(x) + 512) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {- \frac {C_{1} a x^{4}}{4} - \frac {a x^{6}}{8} - \sqrt {- 2 C_{1} - x^{2} + 1} - 1}{2 C_{1} + x^{2}}, \ y{\left (x \right )} = \frac {- \frac {C_{1} a x^{4}}{4} - \frac {a x^{6}}{8} + \sqrt {- 2 C_{1} - x^{2} + 1} - 1}{2 C_{1} + x^{2}}\right ] \]