2.1650   ODE No. 1650

\[ y''(x)-a \sqrt {y'(x)^2+1}=0 \]

✓ Mathematica : cpu = 0.194775 (sec), leaf count = 50

DSolve[-(a*Sqrt[1 + Derivative[1][y][x]^2]) + Derivative[2][y][x] == 0,y[x],x]
 

\[\left \{\left \{y(x)\to c_2-\frac {1}{a \sqrt {\text {sech}^2(a x+c_1)}}\right \},\left \{y(x)\to \frac {1}{a \sqrt {\text {sech}^2(a x+c_1)}}+c_2\right \}\right \}\]

✓ Maple : cpu = 1.203 (sec), leaf count = 16

dsolve(diff(diff(y(x),x),x)-a*(diff(y(x),x)^2+1)^(1/2)=0,y(x))
 

\[y \left (x \right ) = c_{2} +\frac {\cosh \left (a \left (c_{1} +x \right )\right )}{a}\]