54.2.391 problem 970

Internal problem ID [12265]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 970
Date solved : Wednesday, October 01, 2025 at 01:26:20 AM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=-\frac {216 y \left (-2 y^{4}-3 y^{3}-6 y^{2}-6 y+6 x +6\right )}{216 x^{3}+72 y^{8} x +216 y^{7} x +1080 y^{5} x -216 x^{2} y^{4}+594 x y^{6}-648 x y^{3}-1296 x y-324 x^{2} y^{3}-648 x^{2} y^{2}-432 x y^{4}-648 x^{2} y-1944 x y^{2}-36 y^{11}-126 y^{10}-315 y^{9}-8 y^{12}+2808 y^{4}+1728 y^{3}-1296 y^{2}-18 y^{8}+594 y^{7}+4428 y^{5}+2484 y^{6}-1296 y} \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 181

ode:=diff(y(x),x) = -216*y(x)*(-2*y(x)^4-3*y(x)^3-6*y(x)^2-6*y(x)+6*x+6)/(2808*y(x)^4+1728*y(x)^3-648*x^2*y(x)+2484*y(x)^6-1944*x*y(x)^2-432*x*y(x)^4-216*y(x)^4*x^2-126*y(x)^10-8*y(x)^12-36*y(x)^11-648*x*y(x)^3-1296*x*y(x)-1296*y(x)^2+4428*y(x)^5+594*y(x)^7-18*y(x)^8-315*y(x)^9+72*y(x)^8*x+216*y(x)^7*x+1080*y(x)^5*x+594*x*y(x)^6-324*x^2*y(x)^3-648*x^2*y(x)^2+216*x^3-1296*y(x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \frac {-6 \sqrt {3 \ln \left (y\right )-108 c_1 +9}+\left (2 y^{4}+3 y^{3}+6 y^{2}-6 x +6 y\right ) \ln \left (y\right )-72 c_1 y^{4}-108 c_1 y^{3}-216 c_1 y^{2}+216 c_1 x -216 c_1 y+18}{216 c_1 -6 \ln \left (y\right )} &= 0 \\ \frac {6 \sqrt {3 \ln \left (y\right )-108 c_1 +9}+\left (2 y^{4}+3 y^{3}+6 y^{2}-6 x +6 y\right ) \ln \left (y\right )-72 c_1 y^{4}-108 c_1 y^{3}-216 c_1 y^{2}+216 c_1 x -216 c_1 y+18}{216 c_1 -6 \ln \left (y\right )} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.283 (sec). Leaf size: 66

ode=D[y[x],x] == (-216*y[x]*(6 + 6*x - 6*y[x] - 6*y[x]^2 - 3*y[x]^3 - 2*y[x]^4))/(216*x^3 - 1296*y[x] - 1296*x*y[x] - 648*x^2*y[x] - 1296*y[x]^2 - 1944*x*y[x]^2 - 648*x^2*y[x]^2 + 1728*y[x]^3 - 648*x*y[x]^3 - 324*x^2*y[x]^3 + 2808*y[x]^4 - 432*x*y[x]^4 - 216*x^2*y[x]^4 + 4428*y[x]^5 + 1080*x*y[x]^5 + 2484*y[x]^6 + 594*x*y[x]^6 + 594*y[x]^7 + 216*x*y[x]^7 - 18*y[x]^8 + 72*x*y[x]^8 - 315*y[x]^9 - 126*y[x]^10 - 36*y[x]^11 - 8*y[x]^12); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {36 \left (2 y(x)^4+3 y(x)^3+6 y(x)^2+6 y(x)-6 x-3\right )}{\left (y(x) \left (2 y(x)^3+3 y(x)^2+6 y(x)+6\right )-6 x\right )^2}+\log (y(x))=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1296*x - 432*y(x)**4 - 648*y(x)**3 - 1296*y(x)**2 - 1296*y(x) + 1296)*y(x)/(216*x**3 - 216*x**2*y(x)**4 - 324*x**2*y(x)**3 - 648*x**2*y(x)**2 - 648*x**2*y(x) + 72*x*y(x)**8 + 216*x*y(x)**7 + 594*x*y(x)**6 + 1080*x*y(x)**5 - 432*x*y(x)**4 - 648*x*y(x)**3 - 1944*x*y(x)**2 - 1296*x*y(x) - 8*y(x)**12 - 36*y(x)**11 - 126*y(x)**10 - 315*y(x)**9 - 18*y(x)**8 + 594*y(x)**7 + 2484*y(x)**6 + 4428*y(x)**5 + 2808*y(x)**4 + 1728*y(x)**3 - 1296*y(x)**2 - 1296*y(x)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -216*(6*x - 2*y(x)**4 - 3*y(x)**3 - 6*y(x)**2 - 6*y(x) + 6)*y(x)/(-216*x**3 + 216*x**2*y(x)**4 + 324*x**2*y(x)**3 + 648*x**2*y(x)**2 + 648*x**2*y(x) - 72*x*y(x)**8 - 216*x*y(x)**7 - 594*x*y(x)**6 - 1080*x*y(x)**5 + 432*x*y(x)**4 + 648*x*y(x)**3 + 1944*x*y(x)**2 + 1296*x*y(x) + 8*y(x)**12 + 36*y(x)**11 + 126*y(x)**10 + 315*y(x)**9 + 18*y(x)**8 - 594*y(x)**7 - 2484*y(x)**6 - 4428*y(x)**5 - 2808*y(x)**4 - 1728*y(x)**3 + 1296*y(x)**2 + 1296*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method