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 = 42.3344 (sec), leaf count = 352

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

Maple
cpu = 0.056 (sec), leaf count = 27

\[ \left \{ x-\int ^{y \relax (x ) }\! \left (a+b{\it \_a}+\sqrt {{\it B0}\,{\it \_a}+{\it A0}} \right ) ^{-1}{d{\it \_a}}+{\it \_C1}=0 \right \} \] 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*ArcTan[(2*a*b - B0 + 2*b^2*#1)/Sqrt[-4*A0*b^2 +
 (4*a*b - B0)*B0]])/Sqrt[-4*A0*b^2 + (4*a*b - B0)*B0] - (2*Sqrt[2]*Sqrt[2*A0*b^2
 + B0*(-2*a*b + B0 - Sqrt[4*A0*b^2 - 4*a*b*B0 + B0^2])]*ArcTanh[(Sqrt[2]*b*Sqrt[
A0 + B0*#1])/Sqrt[2*A0*b^2 + B0*(-2*a*b + B0 - Sqrt[4*A0*b^2 - 4*a*b*B0 + B0^2])
]])/Sqrt[4*A0*b^2 + B0*(-4*a*b + B0)] + (2*Sqrt[2]*Sqrt[2*A0*b^2 + B0*(-2*a*b + 
B0 + Sqrt[4*A0*b^2 - 4*a*b*B0 + B0^2])]*ArcTanh[(Sqrt[2]*b*Sqrt[A0 + B0*#1])/Sqr
t[2*A0*b^2 + B0*(-2*a*b + B0 + Sqrt[4*A0*b^2 - 4*a*b*B0 + B0^2])]])/Sqrt[4*A0*b^
2 + B0*(-4*a*b + B0)] + Log[a^2 - A0 + 2*a*b*#1 - B0*#1 + b^2*#1^2])/(2*b) & ][x
 + C[1]]}}

Maple raw input

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

Maple raw output

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