Internal problem ID [8139]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 559.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {y \sqrt {\left (y^{\prime }\right )^{2}+1}-a y y^{\prime }-a x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.613 (sec). Leaf size: 392
dsolve(y(x)*(diff(y(x),x)^2+1)^(1/2)-a*y(x)*diff(y(x),x)-a*x=0,y(x), singsol=all)
\begin{align*} x -{\mathrm e}^{\int _{}^{\frac {-a^{2} x +\sqrt {a^{2} y \relax (x )^{2}+a^{2} x^{2}-y \relax (x )^{2}}}{\left (a^{2}-1\right ) y \relax (x )}}\frac {\left (a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a} \right ) a}{\sqrt {\textit {\_a}^{2}+1}\, \left (-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+1}\right ) \left (-a \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} -a \right )}d \textit {\_a}} c_{1} = 0 \\ x -{\mathrm e}^{\int _{}^{-\frac {a^{2} x +\sqrt {a^{2} y \relax (x )^{2}+a^{2} x^{2}-y \relax (x )^{2}}}{\left (a^{2}-1\right ) y \relax (x )}}\frac {\left (a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a} \right ) a}{\sqrt {\textit {\_a}^{2}+1}\, \left (-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+1}\right ) \left (-a \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} -a \right )}d \textit {\_a}} c_{1} = 0 \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {\left (-\textit {\_a}^{2} a^{2}+\textit {\_a}^{2}-a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{\textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} +c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{\textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} \right )+c_{1}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 7.275 (sec). Leaf size: 251
DSolve[-(a*x) - a*y[x]*y'[x] + y[x]*Sqrt[1 + y'[x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )-2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\ y(x)\to \frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )-2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\ y(x)\to -\frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )+2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\ y(x)\to \frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )+2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\ \end{align*}