ODE
\[ y''(x)^2 \left (a^2-b^2 y(x)^2\right )+y'(x)^2 \left (1-b^2 y'(x)^2\right )+2 b^2 y(x) y'(x)^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 121.338 (sec), leaf count = 81
\[\left \{\left \{y(x)\to \frac {a \left (e^{\frac {\sqrt {-1+b^2 c_1{}^2} (x+c_2)}{a}}-c_1\right )}{\sqrt {-1+b^2 c_1{}^2}}\right \},\left \{y(x)\to c_1 e^{c_2 x}-\frac {\sqrt {a^2+\frac {1}{c_2{}^2}}}{b}\right \}\right \}\]
Maple ✓
cpu = 21.924 (sec), leaf count = 162
\[\left [y \left (x \right ) = \frac {\tan \left (\frac {\sqrt {b^{2}}\, \left (\textit {\_C1} -x \right )}{a b}\right ) a}{\sqrt {\tan ^{2}\left (\frac {\sqrt {b^{2}}\, \left (\textit {\_C1} -x \right )}{a b}\right )+1}\, b}, y \left (x \right ) = -\frac {\tan \left (\frac {\sqrt {b^{2}}\, \left (\textit {\_C1} -x \right )}{a b}\right ) a}{\sqrt {\tan ^{2}\left (\frac {\sqrt {b^{2}}\, \left (\textit {\_C1} -x \right )}{a b}\right )+1}\, b}, y \left (x \right ) = \frac {a}{b}, y \left (x \right ) = -\frac {a}{b}, y \left (x \right ) = \textit {\_C1}, y \left (x \right ) = \frac {a \left ({\mathrm e}^{\frac {\sqrt {\textit {\_C1}^{2} b^{2}-1}\, \left (x +\textit {\_C2} \right )}{a}}-\textit {\_C1} \right )}{\sqrt {\textit {\_C1}^{2} b^{2}-1}}\right ]\] Mathematica raw input
DSolve[y'[x]^2*(1 - b^2*y'[x]^2) + 2*b^2*y[x]*y'[x]^2*y''[x] + (a^2 - b^2*y[x]^2)*y''[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> (a*(E^((Sqrt[-1 + b^2*C[1]^2]*(x + C[2]))/a) - C[1]))/Sqrt[-1 + b^2*C[1]^2]}, {y[x] -> E^(x*C[2])*C[1] - Sqrt[a^2 + C[2]^(-2)]/b}}
Maple raw input
dsolve((a^2-b^2*y(x)^2)*diff(diff(y(x),x),x)^2+2*b^2*y(x)*diff(y(x),x)^2*diff(diff(y(x),x),x)+(1-b^2*diff(y(x),x)^2)*diff(y(x),x)^2 = 0, y(x))
Maple raw output
[y(x) = 1/(tan((b^2)^(1/2)*(_C1-x)/a/b)^2+1)^(1/2)*tan((b^2)^(1/2)*(_C1-x)/a/b)*a/b, y(x) = -1/(tan((b^2)^(1/2)*(_C1-x)/a/b)^2+1)^(1/2)*tan((b^2)^(1/2)*(_C1-x)/a/b)*a/b, y(x) = a/b, y(x) = -a/b, y(x) = _C1, y(x) = a*(exp((_C1^2*b^2-1)^(1/2)*(x+_C2)/a)-_C1)/(_C1^2*b^2-1)^(1/2)]