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 = 0.3074 (sec), leaf count = 141
\[\left \{\left \{y(x)\to c_2-i \sqrt {a^2 \left (-1+c_1{}^2\right )-2 a c_1 x+x^2}\right \},\left \{y(x)\to i \sqrt {a^2 \left (-1+c_1{}^2\right )-2 a c_1 x+x^2}+c_2\right \},\left \{y(x)\to c_2-i \sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2}\right \},\left \{y(x)\to i \sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2}+c_2\right \}\right \}\]
Maple ✓
cpu = 0.847 (sec), leaf count = 96
\[\left [y \left (x \right ) = -i x +\textit {\_C1}, y \left (x \right ) = i x +\textit {\_C1}, y \left (x \right ) = -\frac {\left (a +x +\textit {\_C1} \right ) \left (a -x -\textit {\_C1} \right )}{\sqrt {-\textit {\_C1}^{2}-2 \textit {\_C1} x +a^{2}-x^{2}}}+\textit {\_C2}, y \left (x \right ) = \frac {\left (a +x +\textit {\_C1} \right ) \left (a -x -\textit {\_C1} \right )}{\sqrt {-\textit {\_C1}^{2}-2 \textit {\_C1} x +a^{2}-x^{2}}}+\textit {\_C2}\right ]\] 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))
Maple raw output
[y(x) = -I*x+_C1, y(x) = I*x+_C1, y(x) = -(a+x+_C1)*(a-x-_C1)/(-_C1^2-2*_C1*x+a^2-x^2)^(1/2)+_C2, y(x) = (a+x+_C1)*(a-x-_C1)/(-_C1^2-2*_C1*x+a^2-x^2)^(1/2)+_C2]