56.1.79 problem 78

Internal problem ID [8791]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 78
Date solved : Monday, January 27, 2025 at 04:58:47 PM
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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

DSolve[{D[y[x],x]*y[x]/(1+1/2*Sqrt[1+(D[y[x],x])^2])==-x,y[0]==3},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*}