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 = 0.06417 (sec), leaf count = 75
\[\left \{\left \{y(x)\to c_2-\frac {i \sqrt {a^2 x^2+2 a c_1 x+c_1^2-1}}{a}\right \},\left \{y(x)\to c_2+\frac {i \sqrt {a^2 x^2+2 a c_1 x+c_1^2-1}}{a}\right \}\right \}\]
Maple ✓
cpu = 0.134 (sec), leaf count = 41
\[ \left \{ y \left ( x \right ) ={\frac {1}{a} \left ( \left ( -1+ \left ( {\it \_C1}+x \right ) ^{2}{a}^{2} \right ) \sqrt {- \left ( -1+ \left ( {\it \_C1}+x \right ) ^{2}{a}^{2} \right ) ^{-1}}+{\it \_C2}\,a \right ) } \right \} \] 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),'implicit')
Maple raw output
y(x) = ((-1+(_C1+x)^2*a^2)*(-1/(-1+(_C1+x)^2*a^2))^(1/2)+_C2*a)/a