50.1.78 problem 78

Internal problem ID [10076]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 78
Date solved : Tuesday, September 30, 2025 at 06:52:06 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 1.232 (sec). Leaf size: 187

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

\begin{align*} y &= -\frac {\sqrt {-x^{2}+c_1}\, \left (2+\sqrt {\frac {c_1}{-x^{2}+c_1}}\right )}{2} \\ y &= \frac {\sqrt {-x^{2}+c_1}\, \left (2+\sqrt {\frac {c_1}{-x^{2}+c_1}}\right )}{2} \\ y &= -\frac {\sqrt {-9 x^{2}+15 c_1 -6 \sqrt {-3 c_1 \,x^{2}+4 c_1^{2}}}}{3} \\ y &= \frac {\sqrt {-9 x^{2}+15 c_1 -6 \sqrt {-3 c_1 \,x^{2}+4 c_1^{2}}}}{3} \\ y &= -\frac {\sqrt {-9 x^{2}+15 c_1 +6 \sqrt {-3 c_1 \,x^{2}+4 c_1^{2}}}}{3} \\ y &= \frac {\sqrt {-9 x^{2}+15 c_1 +6 \sqrt {-3 c_1 \,x^{2}+4 c_1^{2}}}}{3} \\ \end{align*}

✓ Mathematica. Time used: 1.147 (sec). Leaf size: 153

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

\begin{align*} y(x)&\to \frac {1}{3} \left (e^{c_1}-\sqrt {-9 x^2+4 e^{2 c_1}}\right )\\ y(x)&\to \frac {1}{3} \left (\sqrt {-9 x^2+4 e^{2 c_1}}+e^{c_1}\right )\\ y(x)&\to -\sqrt {-x^2+4 e^{2 c_1}}-e^{c_1}\\ y(x)&\to \sqrt {-x^2+4 e^{2 c_1}}-e^{c_1}\\ 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)/(sqrt(Derivative(y(x), x)**2 + 1)/2 + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out