ODE
\[ y''(x)=a \left (y'(x)^2+1\right )^{3/2} \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.379129 (sec), leaf count = 75
\[\left \{\left \{y(x)\to c_2-\frac {i \sqrt {a^2 x^2+2 a c_1 x-1+c_1{}^2}}{a}\right \},\left \{y(x)\to \frac {i \sqrt {a^2 x^2+2 a c_1 x-1+c_1{}^2}}{a}+c_2\right \}\right \}\]
Maple ✓
cpu = 0.492 (sec), leaf count = 55
\[\left [y \left (x \right ) = \frac {\left (\textit {\_C1} a +a x +1\right ) \left (\textit {\_C1} a +a x -1\right ) \sqrt {-\frac {1}{a^{2} \textit {\_C1}^{2}+2 a^{2} \textit {\_C1} x +a^{2} x^{2}-1}}}{a}+\textit {\_C2}\right ]\] Mathematica raw input
DSolve[y''[x] == a*(1 + y'[x]^2)^(3/2),y[x],x]
Mathematica raw output
{{y[x] -> ((-I)*Sqrt[-1 + a^2*x^2 + 2*a*x*C[1] + C[1]^2])/a + C[2]}, {y[x] -> (I*Sqrt[-1 + a^2*x^2 + 2*a*x*C[1] + C[1]^2])/a + C[2]}}
Maple raw input
dsolve(diff(diff(y(x),x),x) = a*(1+diff(y(x),x)^2)^(3/2), y(x))
Maple raw output
[y(x) = (_C1*a+a*x+1)*(_C1*a+a*x-1)/a*(-1/(_C1^2*a^2+2*_C1*a^2*x+a^2*x^2-1))^(1/2)+_C2]