4.3.5 \(y'(x)=y(x) \sqrt {a+b y(x)}\)

ODE
\[ y'(x)=y(x) \sqrt {a+b y(x)} \] ODE Classification

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

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

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

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

\[ \left \{ x+2\,{\frac {1}{\sqrt {a}}{\it Artanh} \left ({\frac {\sqrt {a+by \relax (x ) }}{\sqrt {a}}} \right ) }+{\it \_C1}=0 \right \} \] 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