Internal problem ID [8136]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 556.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_dAlembert]
Solve \begin {gather*} \boxed {\sqrt {\left (y^{\prime }\right )^{2}+1}+x \left (y^{\prime }\right )^{2}+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.595 (sec). Leaf size: 585
dsolve((diff(y(x),x)^2+1)^(1/2)+x*diff(y(x),x)^2+y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -1 \\ \frac {x^{2} c_{1}}{\left (\sqrt {-4 x y \relax (x )+2+2 \sqrt {4 x^{2}-4 x y \relax (x )+1}}-2 x \right )^{2}}+x +\frac {2 x^{2} \left (\sqrt {2}\, \sqrt {\frac {2 x^{2}-2 x y \relax (x )+\sqrt {4 x^{2}-4 x y \relax (x )+1}+1}{x^{2}}}-2 \arcsinh \left (\frac {\sqrt {-4 x y \relax (x )+2+2 \sqrt {4 x^{2}-4 x y \relax (x )+1}}}{2 x}\right )\right )}{\left (\sqrt {-4 x y \relax (x )+2+2 \sqrt {4 x^{2}-4 x y \relax (x )+1}}-2 x \right )^{2}} = 0 \\ \frac {x^{2} c_{1}}{\left (\sqrt {-4 x y \relax (x )+2+2 \sqrt {4 x^{2}-4 x y \relax (x )+1}}+2 x \right )^{2}}+x +\frac {2 x^{2} \left (\sqrt {2}\, \sqrt {\frac {2 x^{2}-2 x y \relax (x )+\sqrt {4 x^{2}-4 x y \relax (x )+1}+1}{x^{2}}}+2 \arcsinh \left (\frac {\sqrt {-4 x y \relax (x )+2+2 \sqrt {4 x^{2}-4 x y \relax (x )+1}}}{2 x}\right )\right )}{\left (\sqrt {-4 x y \relax (x )+2+2 \sqrt {4 x^{2}-4 x y \relax (x )+1}}+2 x \right )^{2}} = 0 \\ \frac {x^{2} c_{1}}{\left (\sqrt {-4 x y \relax (x )-2 \sqrt {4 x^{2}-4 x y \relax (x )+1}+2}-2 x \right )^{2}}+x +\frac {2 x^{2} \left (\sqrt {\frac {4 x^{2}-4 x y \relax (x )-2 \sqrt {4 x^{2}-4 x y \relax (x )+1}+2}{x^{2}}}+2 \arcsinh \left (-\frac {\sqrt {-4 x y \relax (x )-2 \sqrt {4 x^{2}-4 x y \relax (x )+1}+2}}{2 x}\right )\right )}{\left (\sqrt {-4 x y \relax (x )-2 \sqrt {4 x^{2}-4 x y \relax (x )+1}+2}-2 x \right )^{2}} = 0 \\ \frac {x^{2} c_{1}}{\left (\sqrt {-4 x y \relax (x )-2 \sqrt {4 x^{2}-4 x y \relax (x )+1}+2}+2 x \right )^{2}}+x +\frac {2 x^{2} \left (\sqrt {\frac {4 x^{2}-4 x y \relax (x )-2 \sqrt {4 x^{2}-4 x y \relax (x )+1}+2}{x^{2}}}+2 \arcsinh \left (\frac {\sqrt {-4 x y \relax (x )-2 \sqrt {4 x^{2}-4 x y \relax (x )+1}+2}}{2 x}\right )\right )}{\left (\sqrt {-4 x y \relax (x )-2 \sqrt {4 x^{2}-4 x y \relax (x )+1}+2}+2 x \right )^{2}} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 32.917 (sec). Leaf size: 106
DSolve[y[x] + x*y'[x]^2 + Sqrt[1 + y'[x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=\frac {-y(K[1])-\sqrt {K[1]^2+1}}{K[1]^2},y(x)=e^{2 (\log (K[1])-\log (K[1]+1))} \left (\frac {(-2 K[1]-1) \sqrt {K[1]^2+1}}{K[1]^2}+\tanh ^{-1}\left (\frac {K[1]}{\sqrt {K[1]^2+1}}\right )\right )+c_1 e^{2 (\log (K[1])-\log (K[1]+1))}\right \},\{y(x),K[1]\}\right ] \]