2.1.3 Problem 3

Internal problem ID [4079]
Book : Differential equations, Shepley L. Ross, 1964
Section : 2.4, page 55
Problem number : 3
Date solved : Friday, February 21, 2025 at 08:38:54 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

Solve

\begin{align*} y^{2} \left (x^{2}+1\right )+y+\left (2 x y+1\right ) y^{\prime }&=0 \end{align*}

Unknown ode type.

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right )^{2} \left (x^{2}+1\right )+y \left (x \right )+\left (1+2 x y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {y \left (x \right )^{2} \left (x^{2}+1\right )+y \left (x \right )}{1+2 x y \left (x \right )} \end {array} \]

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 dsolve solution

Solving time : 0.010 (sec)
Leaf size : maple_leaf_size

dsolve(y(x)^2*(x^2+1)+y(x)+(2*x*y(x)+1)*diff(y(x),x) = 0,y(x),singsol=all)
 

\[ \text {No solution found} \]

✗Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

DSolve[{(y[x]^2*(x^2+1)+y[x])+(2*x*y[x]+1)*D[y[x],x]==0,{}},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗Sympy solution

Solving time : 0.000 (sec)
Leaf size : 0

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 + 1)*y(x)**2 + (2*x*y(x) + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out