55.26.14 problem 14

Internal problem ID [13761]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.4. Equations Containing Polynomial Functions of y. subsection 1.4.1-2 Abel equations of the first kind.
Problem number : 14
Date solved : Thursday, October 02, 2025 at 07:59:50 AM
CAS classification : [_rational, _Abel]

\begin{align*} x y^{\prime }&=a y^{3}+3 a b \,x^{n} y^{2}-b n \,x^{n}-2 a \,b^{3} x^{3 n} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 119

ode:=x*diff(y(x),x) = a*y(x)^3+3*a*b*x^n*y(x)^2-b*n*x^n-2*a*b^3*x^(3*n); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -x^{n} b \\ y &= -\frac {{\mathrm e}^{-\frac {3 a \,x^{2 n} b^{2}}{2 n}}}{\sqrt {\frac {a \,\operatorname {Ei}_{1}\left (\frac {3 a \,x^{2 n} b^{2}}{n}\right )+c_1 n}{n}}}-x^{n} b \\ y &= \frac {{\mathrm e}^{-\frac {3 a \,x^{2 n} b^{2}}{2 n}}}{\sqrt {\frac {a \,\operatorname {Ei}_{1}\left (\frac {3 a \,x^{2 n} b^{2}}{n}\right )+c_1 n}{n}}}-x^{n} b \\ \end{align*}

✓ Mathematica. Time used: 1.285 (sec). Leaf size: 128

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

\begin{align*} y(x)&\to -b x^n-\frac {e^{-\frac {3 a b^2 x^{2 n}}{2 n}}}{\sqrt {-\frac {a \operatorname {ExpIntegralEi}\left (-\frac {3 a b^2 x^{2 n}}{n}\right )}{n}+c_1}}\\ y(x)&\to -b x^n+\frac {e^{-\frac {3 a b^2 x^{2 n}}{2 n}}}{\sqrt {-\frac {a \operatorname {ExpIntegralEi}\left (-\frac {3 a b^2 x^{2 n}}{n}\right )}{n}+c_1}}\\ y(x)&\to -b x^n \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-2*a*b**3*x**(3*n) + 3*a*b*x**n*y(x)**2 + a*y(x)**3 - b*n*x**n)/x cannot be solved by the factorable group method