83.22.4 problem 4

Internal problem ID [19258]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 01:15:52 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.093 (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 {-x \left (-2 c_{1} +x \right )}\, \sqrt {-\frac {c_{1}^{2}}{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 {-x \left (-2 c_{1} +x \right )}\, \sqrt {-\frac {c_{1}^{2}}{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: 0.279 (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*}