23.4.116 problem 116

Internal problem ID [6418]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 116
Date solved : Tuesday, September 30, 2025 at 02:56:27 PM
CAS classification : [NONE]

\begin{align*} b +a x y-\left (-12 x^{2}+k \,x^{-1+k}\right ) \left (y^{2}+3 y^{\prime }\right )+2 \left (-4 x^{3}+x^{k}\right ) \left (-y^{3}+y y^{\prime }+y^{\prime \prime }\right )&=0 \end{align*}

✗ Maple

ode:=b+a*x*y(x)-(-12*x^2+k*x^(-1+k))*(y(x)^2+3*diff(y(x),x))+2*(-4*x^3+x^k)*(-y(x)^3+y(x)*diff(y(x),x)+diff(diff(y(x),x),x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=b + a*x*y[x] - (-12*x^2 + k*x^(-1 + k))*(y[x]^2 + 3*D[y[x],x]) + 2*(-4*x^3 + x^k)*(-y[x]^3 + y[x]*D[y[x],x] + D[y[x],{x,2}]) == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
k = symbols("k") 
y = Function("y") 
ode = Eq(a*x*y(x) + b + (-8*x**3 + 2*x**k)*(-y(x)**3 + y(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2))) - (k*x**(k - 1) - 12*x**2)*(y(x)**2 + 3*Derivative(y(x), x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (a*x*y(x) + b - k*x**(k - 1)*y(x)**2 + 8*x