54.2.385 problem 964

Internal problem ID [12259]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 964
Date solved : Wednesday, October 01, 2025 at 01:24:26 AM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=-\frac {8 x \left (a -1\right ) \left (a +1\right )}{8-a^{2} y^{6}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}-2 a^{2} y^{4}-2 a^{6} x^{4}+6 a^{4} x^{4}-6 a^{2} x^{4}-4 a^{2} x^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+4 x^{2} y^{2}+y^{6}-8 a^{2}+2 y^{4}+2 x^{4}+x^{6}-8 y+3 a^{4} y^{4} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+4 a^{4} y^{2} x^{2}-6 y^{4} a^{2} x^{2}-8 y^{2} a^{2} x^{2}} \end{align*}

✓ Maple. Time used: 0.265 (sec). Leaf size: 567

ode:=diff(y(x),x) = -8*x*(a-1)*(a+1)/(8+2*y(x)^4+y(x)^6-8*a^2+4*a^4*y(x)^2*x^2+3*y(x)^4*x^2-2*y(x)^4*a^2-9*y(x)^2*a^2*x^4-3*a^6*y(x)^2*x^4+9*y(x)^2*a^4*x^4+3*a^4*y(x)^4*x^2-6*y(x)^4*a^2*x^2-2*a^6*x^4+6*a^4*x^4-a^2*y(x)^6+a^8*x^6-4*a^6*x^6+6*a^4*x^6-4*a^2*x^6+3*x^4*y(x)^2+4*x^2*y(x)^2-6*a^2*x^4-8*y(x)^2*a^2*x^2-8*y(x)+2*x^4+x^6); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 1.714 (sec). Leaf size: 882

ode=D[y[x],x] == (-8*(-1 + a)*(1 + a)*x)/(8 - 8*a^2 + 2*x^4 - 6*a^2*x^4 + 6*a^4*x^4 - 2*a^6*x^4 + x^6 - 4*a^2*x^6 + 6*a^4*x^6 - 4*a^6*x^6 + a^8*x^6 - 8*y[x] + 4*x^2*y[x]^2 - 8*a^2*x^2*y[x]^2 + 4*a^4*x^2*y[x]^2 + 3*x^4*y[x]^2 - 9*a^2*x^4*y[x]^2 + 9*a^4*x^4*y[x]^2 - 3*a^6*x^4*y[x]^2 + 2*y[x]^4 - 2*a^2*y[x]^4 + 3*x^2*y[x]^4 - 6*a^2*x^2*y[x]^4 + 3*a^4*x^2*y[x]^4 + y[x]^6 - a^2*y[x]^6); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x\frac {8 K[1]}{(a-1) (a+1) \left (a^6 K[1]^6-3 a^4 K[1]^6+3 a^2 K[1]^6-K[1]^6-2 a^4 K[1]^4+4 a^2 K[1]^4-3 a^4 y(x)^2 K[1]^4+6 a^2 y(x)^2 K[1]^4-3 y(x)^2 K[1]^4-2 K[1]^4+3 a^2 y(x)^4 K[1]^2-3 y(x)^4 K[1]^2+4 a^2 y(x)^2 K[1]^2-4 y(x)^2 K[1]^2-y(x)^6-2 y(x)^4-8\right )}dK[1]+\int _1^{y(x)}\left (\frac {8 K[2]}{(a-1)^2 (a+1)^2 \left (-a^6 x^6+3 a^4 x^6-3 a^2 x^6+x^6+2 a^4 x^4-4 a^2 x^4+3 a^4 K[2]^2 x^4-6 a^2 K[2]^2 x^4+3 K[2]^2 x^4+2 x^4-3 a^2 K[2]^4 x^2+3 K[2]^4 x^2-4 a^2 K[2]^2 x^2+4 K[2]^2 x^2+K[2]^6+2 K[2]^4+8\right )}-\frac {\int _1^x-\frac {8 K[1] \left (-6 K[2]^5+12 a^2 K[1]^2 K[2]^3-12 K[1]^2 K[2]^3-8 K[2]^3-6 a^4 K[1]^4 K[2]+12 a^2 K[1]^4 K[2]-6 K[1]^4 K[2]+8 a^2 K[1]^2 K[2]-8 K[1]^2 K[2]\right )}{(a-1) (a+1) \left (a^6 K[1]^6-3 a^4 K[1]^6+3 a^2 K[1]^6-K[1]^6-2 a^4 K[1]^4+4 a^2 K[1]^4-3 a^4 K[2]^2 K[1]^4+6 a^2 K[2]^2 K[1]^4-3 K[2]^2 K[1]^4-2 K[1]^4+3 a^2 K[2]^4 K[1]^2-3 K[2]^4 K[1]^2+4 a^2 K[2]^2 K[1]^2-4 K[2]^2 K[1]^2-K[2]^6-2 K[2]^4-8\right )^2}dK[1] a^2-\int _1^x-\frac {8 K[1] \left (-6 K[2]^5+12 a^2 K[1]^2 K[2]^3-12 K[1]^2 K[2]^3-8 K[2]^3-6 a^4 K[1]^4 K[2]+12 a^2 K[1]^4 K[2]-6 K[1]^4 K[2]+8 a^2 K[1]^2 K[2]-8 K[1]^2 K[2]\right )}{(a-1) (a+1) \left (a^6 K[1]^6-3 a^4 K[1]^6+3 a^2 K[1]^6-K[1]^6-2 a^4 K[1]^4+4 a^2 K[1]^4-3 a^4 K[2]^2 K[1]^4+6 a^2 K[2]^2 K[1]^4-3 K[2]^2 K[1]^4-2 K[1]^4+3 a^2 K[2]^4 K[1]^2-3 K[2]^4 K[1]^2+4 a^2 K[2]^2 K[1]^2-4 K[2]^2 K[1]^2-K[2]^6-2 K[2]^4-8\right )^2}dK[1]-1}{(a-1) (a+1)}\right )dK[2]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(8*x*(a - 1)*(a + 1)/(a**8*x**6 - 4*a**6*x**6 - 3*a**6*x**4*y(x)**2 - 2*a**6*x**4 + 6*a**4*x**6 + 9*a**4*x**4*y(x)**2 + 6*a**4*x**4 + 3*a**4*x**2*y(x)**4 + 4*a**4*x**2*y(x)**2 - 4*a**2*x**6 - 9*a**2*x**4*y(x)**2 - 6*a**2*x**4 - 6*a**2*x**2*y(x)**4 - 8*a**2*x**2*y(x)**2 - a**2*y(x)**6 - 2*a**2*y(x)**4 - 8*a**2 + x**6 + 3*x**4*y(x)**2 + 2*x**4 + 3*x**2*y(x)**4 + 4*x**2*y(x)**2 + y(x)**6 + 2*y(x)**4 - 8*y(x) + 8) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out