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 = 0.183456 (sec), leaf count = 88
\[\left \{\left \{y(x)\to \frac {1}{2} e^{-e^{c_1} \left (c_2+x\right )} \left (e^{2 e^{c_1} \left (c_2+x\right )-2 c_1}-a^2\right )\right \},\left \{y(x)\to \frac {1}{2} e^{-e^{c_1} \left (c_2+x\right )-2 c_1}-\frac {1}{2} a^2 e^{e^{c_1} \left (c_2+x\right )}\right \}\right \}\]
Maple ✓
cpu = 0.102 (sec), leaf count = 68
\[ \left \{ -\ln \left ( {\frac {y \left ( x \right ) }{{\it \_C1}}}+\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{{{\it \_C1}}^{2}}}+{a}^{2}} \right ) {\it \_C1}-x-{\it \_C2}=0,\ln \left ( {\frac {y \left ( x \right ) }{{\it \_C1}}}+\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{{{\it \_C1}}^{2}}}+{a}^{2}} \right ) {\it \_C1}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y[x]*y''[x] == -a^2 + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> (-a^2 + E^(-2*C[1] + 2*E^C[1]*(x + C[2])))/(2*E^(E^C[1]*(x + C[2])))}, {y[x] -> -(a^2*E^(E^C[1]*(x + C[2])))/2 + E^(-2*C[1] - 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),'implicit')
Maple raw output
-ln(y(x)/_C1+(y(x)^2/_C1^2+a^2)^(1/2))*_C1-x-_C2 = 0, ln(y(x)/_C1+(y(x)^2/_C1^2+a^2)^(1/2))*_C1-x-_C2 = 0