ODE
\[ y''(x)^2=a+b y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.155287 (sec), leaf count = 101
\[\left \{\left \{y(x)\to c_2-\frac {e^{-\sqrt {b} \left (c_1+x\right )} \left (a b e^{2 \sqrt {b} x}+e^{2 \sqrt {b} c_1}\right )}{2 b^{3/2}}\right \},\left \{y(x)\to \frac {e^{-\sqrt {b} \left (c_1+x\right )} \left (a b+e^{2 \sqrt {b} \left (c_1+x\right )}\right )}{2 b^{3/2}}+c_2\right \}\right \}\]
Maple ✓
cpu = 0.293 (sec), leaf count = 92
\[ \left \{ y \left ( x \right ) =-{\frac {x}{b}\sqrt {-ab}}+{\it \_C1},y \left ( x \right ) ={\frac {x}{b}\sqrt {-ab}}+{\it \_C1},y \left ( x \right ) ={\it \_C1}+{\it \_C2}\,{{\rm e}^{\sqrt {b}x}}+{\frac {a}{4\,{b}^{2}{\it \_C2}}{{\rm e}^{-\sqrt {b}x}}},y \left ( x \right ) ={\it \_C1}+{\frac {a}{4\,{b}^{2}{\it \_C2}}{{\rm e}^{\sqrt {b}x}}}+{\it \_C2}\,{{\rm e}^{-\sqrt {b}x}} \right \} \] Mathematica raw input
DSolve[y''[x]^2 == a + b*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -(a*b*E^(2*Sqrt[b]*x) + E^(2*Sqrt[b]*C[1]))/(2*b^(3/2)*E^(Sqrt[b]*(x + C[1]))) + C[2]}, {y[x] -> (a*b + E^(2*Sqrt[b]*(x + C[1])))/(2*b^(3/2)*E^(Sqrt[b]*(x + C[1]))) + C[2]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)^2 = a+b*diff(y(x),x)^2, y(x),'implicit')
Maple raw output
y(x) = -1/b*(-a*b)^(1/2)*x+_C1, y(x) = 1/b*(-a*b)^(1/2)*x+_C1, y(x) = _C1+_C2*exp(b^(1/2)*x)+1/4*a/b^2/_C2*exp(-b^(1/2)*x), y(x) = _C1+1/4*a/b^2/_C2*exp(b^(1/2)*x)+_C2*exp(-b^(1/2)*x)