2.5.23 Problem 23

Internal problem ID [8984]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 23
Date solved : Wednesday, March 05, 2025 at 07:13:46 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

Solve

\begin{align*} y^{\prime }&=\frac {x y+3 x -2 y+6}{x y-3 x -2 y+6} \end{align*}

Unknown ode type.

✗ Maple

ode:=diff(y(x),x) = (x*y(x)+3*x-2*y(x)+6)/(x*y(x)-3*x-2*y(x)+6); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
trying Abel 
Looking for potential symmetries 
Looking for potential symmetries 
Looking for potential symmetries 
trying inverse_Riccati 
trying an equivalence to an Abel ODE 
differential order: 1; trying a linearization to 2nd order 
--- trying a change of variables {x -> y(x), y(x) -> x} 
differential order: 1; trying a linearization to 2nd order 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = 4 
`, `-> Computing symmetries using: way = 2 
trying symmetry patterns for 1st order ODEs 
-> trying a symmetry pattern of the form [F(x)*G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)*G(y)] 
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
-> trying a symmetry pattern of the form [F(x),G(x)] 
-> trying a symmetry pattern of the form [F(y),G(y)] 
-> trying a symmetry pattern of the form [F(x)+G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)+G(y)] 
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
-> trying a symmetry pattern of conformal type`
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {x y \left (x \right )+3 x -2 y \left (x \right )+6}{x y \left (x \right )-3 x -2 y \left (x \right )+6} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {x y \left (x \right )+3 x -2 y \left (x \right )+6}{x y \left (x \right )-3 x -2 y \left (x \right )+6} \end {array} \]

✗ Mathematica

ode=D[y[x],x]==(x*y[x]+3*x-2*y[x]+6)/(x*y[x]-3*x-2*y[x]+6); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x*y(x) + 3*x - 2*y(x) + 6)/(x*y(x) - 3*x - 2*y(x) + 6),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out