60.7.184 problem 1804 (book 6.213)

Internal problem ID [11734]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1804 (book 6.213)
Date solved : Sunday, March 30, 2025 at 08:45:02 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) y^{\prime \prime }-\left (\left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right )+\left (y-b \right ) \left (y-c \right )\right ) {y^{\prime }}^{2}+\left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2} \left (A_{0} +\frac {B_{0}}{\left (y-a \right )^{2}}+\frac {C_{1}}{\left (y-b \right )^{2}}+\frac {D_{0}}{\left (y-c \right )^{2}}\right )&=0 \end{align*}

✓ Maple. Time used: 0.261 (sec). Leaf size: 1097

ode:=2*(y(x)-a)*(y(x)-b)*(y(x)-c)*diff(diff(y(x),x),x)-((y(x)-a)^2*(y(x)-b)*(y(x)-c)+(y(x)-c)*(y(x)-b))*diff(y(x),x)^2+(y(x)-a)^2*(y(x)-b)^2*(y(x)-c)^2*(A__0+B__0/(y(x)-a)^2+C__1/(y(x)-b)^2+D__0/(y(x)-c)^2) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 26.168 (sec). Leaf size: 5162

ode=2*(y[x]-a)*(y[x]-b)*(y[x]-c)*D[y[x],{x,2}]-( (y[x]-a)*(y[x]-b)*(y[x]-a)*(y[x]-c)+(y[x]-b)*(y[x]-c) )*D[y[x],x]^2+( (y[x]-a)*(y[x]-b)*(y[x]-c) )^2*(A0+B0/(y[x]-a)^2+C1/(y[x]-b)^2+D0/(y[x]-c)^2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
A__0 = symbols("A__0") 
B__0 = symbols("B__0") 
C__1 = symbols("C__1") 
D__0 = symbols("D__0") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq((-2*a + 2*y(x))*(-b + y(x))*(-c + y(x))*Derivative(y(x), (x, 2)) + (-a + y(x))**2*(-b + y(x))**2*(-c + y(x))**2*(A__0 + B__0/(-a + y(x))**2 + C__1/(-b + y(x))**2 + D__0/(-c + y(x))**2) - ((-a + y(x))**2*(-b + y(x))*(-c + y(x)) + (-b + y(x))*(-c + y(x)))*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt((A__0*a**2*b**2*c**2 - 2*A__0*a**2*b**2*c*y(x) + A__0*a**2*b**2*y(x)**2 - 2*A__0*a**2*b*c**2*y(x) + 4*A__0*a**2*b*c*y(x)**2 - 2*A__0*a**2*b*y(x)**3 + A__0*a**2*c**2*y(x)**2 - 2*A__0*a**2*c*y(x)**3 + A__0*a**2*y(x)**4 - 2*A__0*a*b**2*c**2*y(x) + 4*A__0*a*b**2*c*y(x)**2 - 2*A__0*a*b**2*y(x)**3 + 4*A__0*a*b*c**2*y(x)**2 - 8*A__0*a*b*c*y(x)**3 + 4*A__0*a*b*y(x)**4 - 2*A__0*a*c**2*y(x)**3 + 4*A__0*a*c*y(x)**4 - 2*A__0*a*y(x)**5 + A__0*b**2*c**2*y(x)**2 - 2*A__0*b**2*c*y(x)**3 + A__0*b**2*y(x)**4 - 2*A__0*b*c**2*y(x)**3 + 4*A__0*b*c*y(x)**4 - 2*A__0*b*y(x)**5 + A__0*c**2*y(x)**4 - 2*A__0*c*y(x)**5 + A__0*y(x)**6 + B__0*b**2*c**2 - 2*B__0*b**2*c*y(x) + B__0*b**2*y(x)**2 - 2*B__0*b*c**2*y(x) + 4*B__0*b*c*y(x)**2 - 2*B__0*b*y(x)**3 + B__0*c**2*y(x)**2 - 2*B__0*c*y(x)**3 + B__0*y(x)**4 + C__1*a**2*c**2 - 2*C__1*a**2*c*y(x) + C__1*a**2*y(x)**2 - 2*C__1*a*c**2*y(x) + 4*C__1*a*c*y(x)**2 - 2*C__1*a*y(x)**3 + C__1*c**2*y(x)**2 - 2*C__1*c*y(x)**3 + C__1*y(x)**4 + D__0*a**2*b**2 - 2*D__0*a**2*b*y(x) + D__0*a**2*y(x)**2 - 2*D__0*a*b**2*y(x) + 4*D__0*a*b*y(x)**2 - 2*D__0*a*y(x)**3 + D__0*b**2*y(x)**2 - 2*D__0*b*y(x)**3 + D__0*y(x)**4 - 2*a*b*c*Derivative(y(x), (x, 2)) + 2*a*b*y(x)*Derivative(y(x), (x, 2)) + 2*a*c*y(x)*Derivative(y(x), (x, 2)) - 2*a*y(x)**2*Derivative(y(x), (x, 2)) + 2*b*c*y(x)*Derivative(y(x), (x, 2)) - 2*b*y(x)**2*Derivative(y(x), (x, 2)) - 2*c*y(x)**2*Derivative(y(x), (x, 2)) + 2*y(x)**3*Derivative(y(x), (x, 2)))/((b - y(x))*(c - y(x))*(a**2 - 2*a*y(x) + y(x)**2 + 1))) + Derivative(y(x), x) cannot be solved by the factorable group method