ODE
\[ a^2 y''(x)^2=\left (y'(x)^2+1\right )^3 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.132494 (sec), leaf count = 141
\[\left \{\left \{y(x)\to c_2-i \sqrt {a^2 \left (c_1^2-1\right )-2 a c_1 x+x^2}\right \},\left \{y(x)\to c_2+i \sqrt {a^2 \left (c_1^2-1\right )-2 a c_1 x+x^2}\right \},\left \{y(x)\to c_2-i \sqrt {a^2 \left (c_1^2-1\right )+2 a c_1 x+x^2}\right \},\left \{y(x)\to c_2+i \sqrt {a^2 \left (c_1^2-1\right )+2 a c_1 x+x^2}\right \}\right \}\]
Maple ✓
cpu = 0.191 (sec), leaf count = 78
\[ \left \{ y \left ( x \right ) =-{ \left ( x+{\it \_C1}+a \right ) \left ( -x-{\it \_C1}+a \right ) {\frac {1}{\sqrt {-{{\it \_C1}}^{2}-2\,{\it \_C1}\,x+{a}^{2}-{x}^{2}}}}}+{\it \_C2},y \left ( x \right ) ={ \left ( x+{\it \_C1}+a \right ) \left ( -x-{\it \_C1}+a \right ) {\frac {1}{\sqrt {-{{\it \_C1}}^{2}-2\,{\it \_C1}\,x+{a}^{2}-{x}^{2}}}}}+{\it \_C2} \right \} \] Mathematica raw input
DSolve[a^2*y''[x]^2 == (1 + y'[x]^2)^3,y[x],x]
Mathematica raw output
{{y[x] -> (-I)*Sqrt[x^2 - 2*a*x*C[1] + a^2*(-1 + C[1]^2)] + C[2]}, {y[x] -> I*Sqrt[x^2 - 2*a*x*C[1] + a^2*(-1 + C[1]^2)] + C[2]}, {y[x] -> (-I)*Sqrt[x^2 + 2*a*x*C[1] + a^2*(-1 + C[1]^2)] + C[2]}, {y[x] -> I*Sqrt[x^2 + 2*a*x*C[1] + a^2*(-1 + C[1]^2)] + C[2]}}
Maple raw input
dsolve(a^2*diff(diff(y(x),x),x)^2 = (1+diff(y(x),x)^2)^3, y(x),'implicit')
Maple raw output
y(x) = -(x+_C1+a)*(-x-_C1+a)/(-_C1^2-2*_C1*x+a^2-x^2)^(1/2)+_C2, y(x) = (x+_C1+a)*(-x-_C1+a)/(-_C1^2-2*_C1*x+a^2-x^2)^(1/2)+_C2