60.1.554 problem 557

Internal problem ID [10568]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 557
Date solved : Monday, January 27, 2025 at 09:06:35 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.144 (sec). Leaf size: 97

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

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

✓ Solution by Mathematica

Time used: 0.299 (sec). Leaf size: 37

DSolve[-y[x] + x*(D[y[x],x] + Sqrt[1 + D[y[x],x]^2])==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {-x (x-c_1)} \\ y(x)\to \sqrt {-x (x-c_1)} \\ \end{align*}