ODE
\[ y''(x)^2=a+b y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.510556 (sec), leaf count = 155
\[\left \{\left \{y(x)\to c_2-\frac {\sqrt {a}}{\sqrt {b} \sqrt {b \text {sech}^2\left (\sqrt {b} (x-c_1)\right )}}\right \},\left \{y(x)\to \frac {\sqrt {a}}{\sqrt {b} \sqrt {b \text {sech}^2\left (\sqrt {b} (x-c_1)\right )}}+c_2\right \},\left \{y(x)\to c_2-\frac {\sqrt {a}}{\sqrt {b} \sqrt {b \text {sech}^2\left (\sqrt {b} (x+c_1)\right )}}\right \},\left \{y(x)\to \frac {\sqrt {a}}{\sqrt {b} \sqrt {b \text {sech}^2\left (\sqrt {b} (x+c_1)\right )}}+c_2\right \}\right \}\]
Maple ✓
cpu = 5.308 (sec), leaf count = 92
\[\left [y \left (x \right ) = -\frac {\sqrt {-a b}\, x}{b}+\textit {\_C1}, y \left (x \right ) = \frac {\sqrt {-a b}\, x}{b}+\textit {\_C1}, y \left (x \right ) = \textit {\_C1} +\textit {\_C2} \,{\mathrm e}^{x \sqrt {b}}+\frac {a \,{\mathrm e}^{-x \sqrt {b}}}{4 b^{2} \textit {\_C2}}, y \left (x \right ) = \textit {\_C1} +\frac {a \,{\mathrm e}^{x \sqrt {b}}}{4 b^{2} \textit {\_C2}}+\textit {\_C2} \,{\mathrm e}^{-x \sqrt {b}}\right ]\] Mathematica raw input
DSolve[y''[x]^2 == a + b*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> C[2] - Sqrt[a]/(Sqrt[b]*Sqrt[b*Sech[Sqrt[b]*(x - C[1])]^2])}, {y[x] -> C[2] + Sqrt[a]/(Sqrt[b]*Sqrt[b*Sech[Sqrt[b]*(x - C[1])]^2])}, {y[x] -> C[2] - Sqrt[a]/(Sqrt[b]*Sqrt[b*Sech[Sqrt[b]*(x + C[1])]^2])}, {y[x] -> C[2] + Sqrt[a]/(Sqrt[b]*Sqrt[b*Sech[Sqrt[b]*(x + C[1])]^2])}}
Maple raw input
dsolve(diff(diff(y(x),x),x)^2 = a+b*diff(y(x),x)^2, y(x))
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(x*b^(1/2))+1/4*a/b^2/_C2*exp(-x*b^(1/2)), y(x) = _C1+1/4*a/b^2/_C2*exp(x*b^(1/2))+_C2*exp(-x*b^(1/2))]