ODE
\[ y''(x)=a \sqrt {y'(x)^2+1} \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0332994 (sec), leaf count = 19
\[\left \{\left \{y(x)\to \frac {\cosh \left (a x+c_1\right )}{a}+c_2\right \}\right \}\]
Maple ✓
cpu = 0.384 (sec), leaf count = 16
\[ \left \{ y \left ( x \right ) ={\frac {\cosh \left ( a \left ( {\it \_C1}+x \right ) \right ) }{a}}+{\it \_C2} \right \} \] Mathematica raw input
DSolve[y''[x] == a*Sqrt[1 + y'[x]^2],y[x],x]
Mathematica raw output
{{y[x] -> C[2] + Cosh[a*x + C[1]]/a}}
Maple raw input
dsolve(diff(diff(y(x),x),x) = a*(1+diff(y(x),x)^2)^(1/2), y(x),'implicit')
Maple raw output
y(x) = 1/a*cosh(a*(_C1+x))+_C2