31.6.14 problem 14

Internal problem ID [5763]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 14
Date solved : Monday, January 27, 2025 at 01:13:04 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.183 (sec). Leaf size: 97

dsolve(y(x)=x*diff(y(x),x)+x*sqrt(1+(diff(y(x),x))^2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.274 (sec). Leaf size: 37

DSolve[y[x]==x*D[y[x],x]+x*Sqrt[1+(D[y[x],x])^2],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*}