##### 4.2.50 $$y'(x)=a+\sqrt {\text {A0}+\text {B0} y(x)}+b y(x)$$

ODE
$y'(x)=a+\sqrt {\text {A0}+\text {B0} y(x)}+b y(x)$ ODE Classiﬁcation

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

Mathematica
cpu = 0.525077 (sec), leaf count = 125

$\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {2 \text {B0} \sqrt {-4 a b \text {B0}+4 \text {A0} b^2+\text {B0}^2} \tanh ^{-1}\left (\frac {2 b \sqrt {\text {\#1} \text {B0}+\text {A0}}+\text {B0}}{\sqrt {-4 a b \text {B0}+4 \text {A0} b^2+\text {B0}^2}}\right )}{b \left (\text {B0}^2-4 b (a \text {B0}-\text {A0} b)\right )}+\frac {\log \left (-b (\text {\#1} \text {B0}+\text {A0})-\text {B0} \sqrt {\text {\#1} \text {B0}+\text {A0}}-a \text {B0}+\text {A0} b\right )}{b}\& \right ][x+c_1]\right \}\right \}$

Maple
cpu = 0.034 (sec), leaf count = 27

$\left [x -\left (\int _{}^{y \left (x \right )}\frac {1}{a +b \textit {\_a} +\sqrt {\mathit {B0} \textit {\_a} +\mathit {A0}}}d \textit {\_a} \right )+\textit {\_C1} = 0\right ]$ Mathematica raw input

DSolve[y'[x] == a + b*y[x] + Sqrt[A0 + B0*y[x]],y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[(2*B0*Sqrt[4*A0*b^2 - 4*a*b*B0 + B0^2]*ArcTanh[(B0 + 2
*b*Sqrt[A0 + B0*#1])/Sqrt[4*A0*b^2 - 4*a*b*B0 + B0^2]])/(b*(B0^2 - 4*b*(-(A0*b)
+ a*B0))) + Log[A0*b - a*B0 - B0*Sqrt[A0 + B0*#1] - b*(A0 + B0*#1)]/b & ][x + C[
1]]}}

Maple raw input

dsolve(diff(y(x),x) = a+b*y(x)+(A0+B0*y(x))^(1/2), y(x))

Maple raw output

[x-Intat(1/(a+b*_a+(B0*_a+A0)^(1/2)),_a = y(x))+_C1 = 0]