ODE
\[ y(x) y''(x)=y'(x)^2-a^2 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.442314 (sec), leaf count = 87
\[\left \{\left \{y(x)\to -\frac {i a e^{-c_1} \tanh \left (e^{c_1} (x+c_2)\right )}{\sqrt {-\text {sech}^2\left (e^{c_1} (x+c_2)\right )}}\right \},\left \{y(x)\to \frac {i a e^{-c_1} \tanh \left (e^{c_1} (x+c_2)\right )}{\sqrt {-\text {sech}^2\left (e^{c_1} (x+c_2)\right )}}\right \}\right \}\]
Maple ✓
cpu = 1.868 (sec), leaf count = 87
\[\left [y \left (x \right ) = \frac {\textit {\_C1} \left ({\mathrm e}^{-\frac {2 x}{\textit {\_C1}}} {\mathrm e}^{-\frac {2 \textit {\_C2}}{\textit {\_C1}}}-a^{2}\right ) {\mathrm e}^{\frac {x}{\textit {\_C1}}} {\mathrm e}^{\frac {\textit {\_C2}}{\textit {\_C1}}}}{2}, y \left (x \right ) = \frac {\textit {\_C1} \left ({\mathrm e}^{\frac {2 x}{\textit {\_C1}}} {\mathrm e}^{\frac {2 \textit {\_C2}}{\textit {\_C1}}}-a^{2}\right ) {\mathrm e}^{-\frac {x}{\textit {\_C1}}} {\mathrm e}^{-\frac {\textit {\_C2}}{\textit {\_C1}}}}{2}\right ]\] Mathematica raw input
DSolve[y[x]*y''[x] == -a^2 + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> ((-I)*a*Tanh[E^C[1]*(x + C[2])])/(E^C[1]*Sqrt[-Sech[E^C[1]*(x + C[2])]^2])}, {y[x] -> (I*a*Tanh[E^C[1]*(x + C[2])])/(E^C[1]*Sqrt[-Sech[E^C[1]*(x + C[2])]^2])}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2-a^2, y(x))
Maple raw output
[y(x) = 1/2*_C1*(1/exp(x/_C1)^2/exp(1/_C1*_C2)^2-a^2)*exp(x/_C1)*exp(1/_C1*_C2), y(x) = 1/2*_C1*(exp(x/_C1)^2*exp(1/_C1*_C2)^2-a^2)/exp(x/_C1)/exp(1/_C1*_C2)]