56.1.79 problem 78

Internal problem ID [8791]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 78
Date solved : Wednesday, March 05, 2025 at 06:48:42 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

With initial conditions

\begin{align*} y \left (0\right )&=3 \end{align*}

✓ Maple. Time used: 10.225 (sec). Leaf size: 29

ode:=diff(y(x),x)*y(x)/(1+1/2*(1+diff(y(x),x)^2)^(1/2)) = -x; 
ic:=y(0) = 3; 
dsolve([ode,ic],y(x), singsol=all);
 

\begin{align*} y &= -3+\sqrt {-x^{2}+36} \\ y &= 1+\sqrt {-x^{2}+4} \\ \end{align*}

✓ Mathematica. Time used: 0.573 (sec). Leaf size: 35

ode=D[y[x],x]*y[x]/(1+1/2*Sqrt[1+(D[y[x],x])^2])==-x; 
ic=y[0]==3; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \sqrt {4-x^2}+1 \\ y(x)\to \sqrt {36-x^2}-3 \\ \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 = {y(0): 3} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants