2.1.19 Problem 19

Internal problem ID [8731]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 19
Date solved : Friday, February 21, 2025 at 08:26:01 PM
CAS classification : [`y=_G(x,y')`]

Solve

\begin{align*} y^{\prime }&=\sqrt {1-x^{2}-y^{2}} \end{align*}

Unknown ode type.

Maple step by step solution

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

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying homogeneous types: 
differential order: 1; looking for linear symmetries 
trying exact 
Looking for potential symmetries 
trying an equivalence to an Abel ODE 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = 5 
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(diff(y(x),x) = (1-x^2-y(x)^2)^(1/2),y(x),singsol=all)
 

\[ \text {No solution found} \]

✗Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

DSolve[{D[y[x],x]==Sqrt[ 1-x^2-y[x]^2],{}},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(-sqrt(-x**2 - y(x)**2 + 1) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-x**2 - y(x)**2 + 1) + Derivative(y(x), x) cannot be solved by the lie group method