##### 4.16.13 $$y'(x)^2=a+b y(x)^2$$

ODE
$y'(x)^2=a+b y(x)^2$ ODE Classiﬁcation

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for $$y'$$

Mathematica
cpu = 0.255653 (sec), leaf count = 171

$\left \{\left \{y(x)\to -\frac {\sqrt {a} \tanh \left (\sqrt {b} (x-c_1)\right )}{\sqrt {b \text {sech}^2\left (\sqrt {b} (x-c_1)\right )}}\right \},\left \{y(x)\to \frac {\sqrt {a} \tanh \left (\sqrt {b} (x-c_1)\right )}{\sqrt {b \text {sech}^2\left (\sqrt {b} (x-c_1)\right )}}\right \},\left \{y(x)\to -\frac {\sqrt {a} \tanh \left (\sqrt {b} (x+c_1)\right )}{\sqrt {b \text {sech}^2\left (\sqrt {b} (x+c_1)\right )}}\right \},\left \{y(x)\to \frac {\sqrt {a} \tanh \left (\sqrt {b} (x+c_1)\right )}{\sqrt {b \text {sech}^2\left (\sqrt {b} (x+c_1)\right )}}\right \}\right \}$

Maple
cpu = 0.411 (sec), leaf count = 114

$\left [y \left (x \right ) = \frac {\sqrt {-a b}}{b}, y \left (x \right ) = -\frac {\sqrt {-a b}}{b}, y \left (x \right ) = \frac {\left ({\mathrm e}^{-2 \textit {\_C1} \sqrt {b}} {\mathrm e}^{2 x \sqrt {b}}-a \right ) {\mathrm e}^{\textit {\_C1} \sqrt {b}} {\mathrm e}^{-x \sqrt {b}}}{2 \sqrt {b}}, y \left (x \right ) = \frac {\left ({\mathrm e}^{2 \textit {\_C1} \sqrt {b}} {\mathrm e}^{-2 x \sqrt {b}}-a \right ) {\mathrm e}^{-\textit {\_C1} \sqrt {b}} {\mathrm e}^{x \sqrt {b}}}{2 \sqrt {b}}\right ]$ Mathematica raw input

DSolve[y'[x]^2 == a + b*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -((Sqrt[a]*Tanh[Sqrt[b]*(x - C[1])])/Sqrt[b*Sech[Sqrt[b]*(x - C[1])]^2
])}, {y[x] -> (Sqrt[a]*Tanh[Sqrt[b]*(x - C[1])])/Sqrt[b*Sech[Sqrt[b]*(x - C[1])]
^2]}, {y[x] -> -((Sqrt[a]*Tanh[Sqrt[b]*(x + C[1])])/Sqrt[b*Sech[Sqrt[b]*(x + C[1
])]^2])}, {y[x] -> (Sqrt[a]*Tanh[Sqrt[b]*(x + C[1])])/Sqrt[b*Sech[Sqrt[b]*(x + C
[1])]^2]}}

Maple raw input

dsolve(diff(y(x),x)^2 = a+b*y(x)^2, y(x))

Maple raw output

[y(x) = 1/b*(-a*b)^(1/2), y(x) = -1/b*(-a*b)^(1/2), y(x) = 1/2*(1/exp(_C1*b^(1/2
))^2*exp(x*b^(1/2))^2-a)/b^(1/2)*exp(_C1*b^(1/2))/exp(x*b^(1/2)), y(x) = 1/2*(ex
p(_C1*b^(1/2))^2/exp(x*b^(1/2))^2-a)/b^(1/2)/exp(_C1*b^(1/2))*exp(x*b^(1/2))]