Internal problem ID [8894]
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]
\[ \boxed {y \sqrt {{y^{\prime }}^{2}+1}-a y y^{\prime }=a x} \]
✓ Solution by Maple
Time used: 0.516 (sec). Leaf size: 378
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*} -{\mathrm e}^{a \left (\int _{}^{\frac {-a^{2} x +\sqrt {y \left (x \right )^{2} \left (a^{2}-1\right )+a^{2} x^{2}}}{\left (a^{2}-1\right ) y \left (x \right )}}\frac {a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}\, \left (-\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +a \left (\textit {\_a}^{2}+1\right )\right ) \left (\textit {\_a} a -\sqrt {\textit {\_a}^{2}+1}\right )}d \textit {\_a} \right )} c_{1} +x &= 0 \\ -{\mathrm e}^{a \left (\int _{}^{\frac {-a^{2} x -\sqrt {y \left (x \right )^{2} \left (a^{2}-1\right )+a^{2} x^{2}}}{\left (a^{2}-1\right ) y \left (x \right )}}\frac {a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}\, \left (-\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +a \left (\textit {\_a}^{2}+1\right )\right ) \left (\textit {\_a} a -\sqrt {\textit {\_a}^{2}+1}\right )}d \textit {\_a} \right )} c_{1} +x &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\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}}{\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}\right ) \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\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}}{\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}\right ) \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} \right )+c_{1} \right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 6.694 (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*}