Internal problem ID [3835]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 37
Problem number: 1152.
ODE order: 1.
ODE degree: -1.
CAS Maple gives this as type [_Clairaut]
Solve \begin {gather*} \boxed {y^{\prime } \ln \left (y^{\prime }+\sqrt {a +\left (y^{\prime }\right )^{2}}\right )-\sqrt {\left (y^{\prime }\right )^{2}+1}-y^{\prime } x +y=0} \end {gather*}
✓ Solution by Maple
Time used: 11.062 (sec). Leaf size: 129
dsolve(diff(y(x),x)*ln(diff(y(x),x)+sqrt(a+diff(y(x),x)^2))-sqrt(1+diff(y(x),x)^2)-x*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
\begin{align*} \left [x \left (\textit {\_T} \right ) = \frac {\ln \left (\textit {\_T} +\sqrt {\textit {\_T}^{2}+a}\right ) \sqrt {\textit {\_T}^{2}+a}\, \sqrt {\textit {\_T}^{2}+1}-\textit {\_T} \sqrt {\textit {\_T}^{2}+a}+\textit {\_T} \sqrt {\textit {\_T}^{2}+1}}{\sqrt {\textit {\_T}^{2}+a}\, \sqrt {\textit {\_T}^{2}+1}}, y \left (\textit {\_T} \right ) = \frac {\sqrt {\textit {\_T}^{2}+1}\, \textit {\_T}^{2}+\sqrt {\textit {\_T}^{2}+a}}{\sqrt {\textit {\_T}^{2}+a}\, \sqrt {\textit {\_T}^{2}+1}}\right ] \\ y \relax (x ) = -c_{1} \ln \left (c_{1}+\sqrt {c_{1}^{2}+a}\right )+\sqrt {c_{1}^{2}+1}+c_{1} x \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.044 (sec). Leaf size: 37
DSolve[y'[x]*Log[y'[x]+Sqrt[a+(y'[x])^2]]-Sqrt[1+(y'[x])^2]-x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \left (x-\log \left (\sqrt {a+c_1{}^2}+c_1\right )\right )+\sqrt {1+c_1{}^2} \\ \end{align*}