Internal problem ID [9230]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1652 (book 6.61).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a \sqrt {1+\left (y^{\prime }\right )^{2}}-b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 31
dsolve(diff(diff(y(x),x),x)=a*sqrt(1+diff(y(x),x)^2)+b,y(x), singsol=all)
\[ y \relax (x ) = \int \RootOf \left (x -\left (\int _{}^{\textit {\_Z}}\frac {1}{a \sqrt {\textit {\_f}^{2}+1}+b}d \textit {\_f} \right )+c_{1}\right )d x +c_{2} \]
✓ Solution by Mathematica
Time used: 60.51 (sec). Leaf size: 797
DSolve[y''[x]==a*Sqrt[1+y'[x]^2]+b,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\frac {b \left (\log \left (\text {InverseFunction}\left [\frac {\frac {2 b \text {ArcTan}\left (\frac {b+a \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{a}\&\right ][x+c_1]{}^2 a^2+a^2-b^2\right )+\log \left (\frac {2 a^2 \left (a-i \sqrt {(a-b) (a+b)} \text {InverseFunction}\left [\frac {\frac {2 b \text {ArcTan}\left (\frac {b+a \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{a}\&\right ][x+c_1]+b \sqrt {\text {InverseFunction}\left [\frac {\frac {2 b \text {ArcTan}\left (\frac {b+a \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{a}\&\right ][x+c_1]{}^2+1}\right )}{b^3 \left (a \text {InverseFunction}\left [\frac {\frac {2 b \text {ArcTan}\left (\frac {b+a \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{a}\&\right ][x+c_1]+i \sqrt {(a-b) (a+b)}\right )}\right )+\log \left (\frac {2 a^2 \left (a+i \sqrt {(a-b) (a+b)} \text {InverseFunction}\left [\frac {\frac {2 b \text {ArcTan}\left (\frac {b+a \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{a}\&\right ][x+c_1]+b \sqrt {\text {InverseFunction}\left [\frac {\frac {2 b \text {ArcTan}\left (\frac {b+a \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{a}\&\right ][x+c_1]{}^2+1}\right )}{b^3 \left (a \text {InverseFunction}\left [\frac {\frac {2 b \text {ArcTan}\left (\frac {b+a \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{a}\&\right ][x+c_1]-i \sqrt {(a-b) (a+b)}\right )}\right )\right )}{2 a^2}+\frac {\sqrt {\text {InverseFunction}\left [\frac {\frac {2 b \text {ArcTan}\left (\frac {b+a \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )}{a}\&\right ][x+c_1]{}^2+1}}{a} \\ \end{align*}