##### 4.3.4 $$y'(x)=\sqrt {a+b y(x)^2}$$

ODE
$y'(x)=\sqrt {a+b y(x)^2}$ ODE Classiﬁcation

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

Mathematica
cpu = 0.301869 (sec), leaf count = 82

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

Maple
cpu = 0.011 (sec), leaf count = 29

$\left [x -\frac {\ln \left (y \left (x \right ) \sqrt {b}+\sqrt {a +b y \left (x \right )^{2}}\right )}{\sqrt {b}}+\textit {\_C1} = 0\right ]$ Mathematica raw input

DSolve[y'[x] == Sqrt[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]}}

Maple raw input

dsolve(diff(y(x),x) = (a+b*y(x)^2)^(1/2), y(x))

Maple raw output

[x-ln(y(x)*b^(1/2)+(a+b*y(x)^2)^(1/2))/b^(1/2)+_C1 = 0]