55.25.15 problem 15

Internal problem ID [13724]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.4-2.
Problem number : 15
Date solved : Thursday, October 02, 2025 at 04:33:02 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \left (y x +x^{2}+a \right ) y^{\prime }&=y^{2}+y x +b \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 91

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

\begin{align*} y &= \frac {c_1 a b x +x +\sqrt {\left (a +b \right ) \left (-1+\left (a \,x^{2}+b \,x^{2}+a^{2}\right ) c_1 \right )}}{c_1 \,a^{2}-1} \\ y &= \frac {c_1 a b x +x -\sqrt {\left (a +b \right ) \left (-1+\left (a \,x^{2}+b \,x^{2}+a^{2}\right ) c_1 \right )}}{c_1 \,a^{2}-1} \\ \end{align*}

✓ Mathematica. Time used: 4.063 (sec). Leaf size: 186

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

\begin{align*} y(x)&\to -\frac {-\frac {1}{\frac {a}{a^2+a x^2+b x^2}-\frac {x}{\left (a^2+a x^2+b x^2\right )^{3/2} \sqrt {-\frac {1}{(a+b) \left (a^2+a x^2+b x^2\right )}+c_1}}}+a+x^2}{x}\\ y(x)&\to -\frac {-\frac {1}{\frac {a}{a^2+a x^2+b x^2}+\frac {x}{\left (a^2+a x^2+b x^2\right )^{3/2} \sqrt {-\frac {1}{(a+b) \left (a^2+a x^2+b x^2\right )}+c_1}}}+a+x^2}{x}\\ y(x)&\to \frac {b x}{a} \end{align*}

✓ Sympy. Time used: 53.404 (sec). Leaf size: 37

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

\[ C_{1} + \frac {a \log {\left (- a y{\left (x \right )} + b x \right )}}{a + b} - \frac {a \log {\left (a + b + \left (x + y{\left (x \right )}\right )^{2} \right )}}{2 \left (a + b\right )} = 0 \]