4.3.5 $y^{\prime }(x)=y(x)\sqrt{a+by(x)}$

ODE
$$y^{\prime }(x)=y(x)\sqrt{a+by(x)}$$ ODE Classification

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

Mathematica ✓
cpu = 0.0310508 (sec), leaf count = 27

$$\left\{\{y(x)\to -\frac{a{\text{sech}}^2\left(\frac{1}{2}\sqrt{a}(c_1+x)\right)}{b}\}\right\}$$

Maple ✓
cpu = 0.015 (sec), leaf count = 24

$$\{x+2\frac{1}{\sqrt{a}}\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\frac{\sqrt{a+by\left(x\right)}}{\sqrt{a}}\right)+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

x+2/a^(1/2)*arctanh((a+b*y(x))^(1/2)/a^(1/2))+_C1 = 0