82.18.8 problem Ex. 9

Internal problem ID [18759]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Examples on chapter III. page 38
Problem number : Ex. 9
Date solved : Monday, March 31, 2025 at 06:08:50 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.228 (sec). Leaf size: 33

ode:=(-y(x)+x*diff(y(x),x))^2 = diff(y(x),x)^2-2*y(x)/x*diff(y(x),x)+1; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 2.24 (sec). Leaf size: 161

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

\begin{align*} y(x)\to -x \cos \left (2 \arctan \left (\frac {\left (\sqrt {x-1}-1\right ) \left (\sqrt {3}-\sqrt {x+1}\right )}{\left (\sqrt {x-1}+1\right ) \left (\sqrt {3} \sqrt {x+1}-1\right )-2 x}\right )+i c_1\right ) \\ y(x)\to -x \cos \left (2 \arctan \left (\frac {\left (\sqrt {x-1}-1\right ) \left (\sqrt {3}-\sqrt {x+1}\right )}{\left (\sqrt {x-1}+1\right ) \left (\sqrt {3} \sqrt {x+1}-1\right )-2 x}\right )-i c_1\right ) \\ y(x)\to -x \\ y(x)\to x \\ \end{align*}

✗ Sympy

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

Timed Out