69.1.5 problem 6
Internal
problem
ID
[17955]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
1.
Basic
concepts
and
definitions.
Exercises
page
18
Problem
number
:
6
Date
solved
:
Thursday, October 02, 2025 at 02:31:09 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} y^{\prime }&=\sqrt {x^{2}-y}-x \end{align*}
✓ Maple. Time used: 0.036 (sec). Leaf size: 171
ode:=diff(y(x),x) = (x^2-y(x))^(1/2)-x;
dsolve(ode,y(x), singsol=all);
\[
\frac {250 \left (x^{2}-y\right )^{{3}/{2}} \left (x^{2}+4 y\right ) \left (x^{6} c_1 y^{2}+\frac {12 x^{4} c_1 y^{3}}{5}+\frac {48 x^{2} c_1 y^{4}}{25}+\frac {64 c_1 y^{5}}{125}-\frac {1}{125}\right )-250 \left (x^{6} c_1 y^{2}+\frac {12 x^{4} c_1 y^{3}}{5}+\frac {48 x^{2} c_1 y^{4}}{25}+\frac {64 c_1 y^{5}}{125}+\frac {1}{125}\right ) x \left (x^{4}+\frac {5 y x^{2}}{2}+10 y^{2}\right )}{\left (5 x^{2}+4 y\right )^{3} y^{2} \left (2 \sqrt {x^{2}-y}+3 x \right )^{3} \left (-\sqrt {x^{2}-y}+x \right )^{2}} = 0
\]
✓ Mathematica. Time used: 5.607 (sec). Leaf size: 416
ode=D[y[x],x]==Sqrt[x^2-y[x]]-x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,1\right ]\\ y(x)&\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,2\right ]\\ y(x)&\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,3\right ]\\ y(x)&\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,4\right ]\\ y(x)&\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,5\right ]\\ y(x)&\to 0 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x - sqrt(x**2 - y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE x - sqrt(x**2 - y(x)) + Derivative(y(x), x) cannot be solved by the lie group method