56.16.6 problem Ex 6

Internal problem ID [14178]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 27. Clairaut equation. Page 56
Problem number : Ex 6
Date solved : Thursday, October 02, 2025 at 09:22:41 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 103

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

\begin{align*} y &= 0 \\ y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} -1\right )^{2}}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_1 \right ) x \\ y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} -1\right )^{2}}-2 \textit {\_a}^{2}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_1 \right ) x \\ \end{align*}

✓ Mathematica. Time used: 2.312 (sec). Leaf size: 167

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

\begin{align*} y(x)&\to -\sqrt {-x \left (x+2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}}\\ y(x)&\to \sqrt {-x \left (x+2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}}\\ y(x)&\to e^{\frac {c_1}{2}}-\sqrt {x \left (-x+2 e^{\frac {c_1}{2}}\right )}\\ y(x)&\to \sqrt {x \left (-x+2 e^{\frac {c_1}{2}}\right )}+e^{\frac {c_1}{2}}\\ y(x)&\to -\sqrt {-x^2}\\ y(x)&\to \sqrt {-x^2} \end{align*}

✗ Sympy

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

Timed Out