ODE
\[ \left (x y''(x)-y'(x)\right )^2=y''(x)^2+1 \] ODE Classification
[[_2nd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.198335 (sec), leaf count = 58
\[\left \{\left \{y(x)\to \frac {c_1 x^2}{2}-\sqrt {1+c_1{}^2} x+c_2\right \},\left \{y(x)\to \frac {c_1 x^2}{2}+\sqrt {1+c_1{}^2} x+c_2\right \}\right \}\]
Maple ✓
cpu = 0.487 (sec), leaf count = 63
\[\left [y \left (x \right ) = \frac {x \sqrt {-x^{2}+1}}{2}+\frac {\arcsin \left (x \right )}{2}+\textit {\_C1}, y \left (x \right ) = -\frac {x \sqrt {-x^{2}+1}}{2}-\frac {\arcsin \left (x \right )}{2}+\textit {\_C1}, y \left (x \right ) = \frac {x^{2} \sqrt {\textit {\_C1}^{2}-1}}{2}+\textit {\_C1} x +\textit {\_C2}\right ]\] Mathematica raw input
DSolve[(-y'[x] + x*y''[x])^2 == 1 + y''[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> (x^2*C[1])/2 - x*Sqrt[1 + C[1]^2] + C[2]}, {y[x] -> (x^2*C[1])/2 + x*Sqrt[1 + C[1]^2] + C[2]}}
Maple raw input
dsolve((x*diff(diff(y(x),x),x)-diff(y(x),x))^2 = 1+diff(diff(y(x),x),x)^2, y(x))
Maple raw output
[y(x) = 1/2*x*(-x^2+1)^(1/2)+1/2*arcsin(x)+_C1, y(x) = -1/2*x*(-x^2+1)^(1/2)-1/2*arcsin(x)+_C1, y(x) = 1/2*x^2*(_C1^2-1)^(1/2)+_C1*x+_C2]