4.3.1 $y^{\prime }(x)=ax+b\sqrt{y(x)}$

ODE
$$y^{\prime }(x)=ax+b\sqrt{y(x)}$$ ODE Classification

[[_homogeneous, `class G`], _Chini]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 0.221863 (sec), leaf count = 116

$$\text{Solve}[\frac{b^2\log \left(b^2(-\sqrt{\frac{b^{2}y(x)}{a^2{x}^2}}+\frac{2y(x)}{a{x}^2}-1)\right)+\frac{2b^3{\tanh }^{-1}\left(\frac{b^2-4a\sqrt{\frac{b^{2}y(x)}{a^2{x}^2}}}{b\sqrt{8a+b^2}}\right)}{\sqrt{8a+b^2}}+2a{c}_1+2b^2\log (x)}{a}=0,y(x)]$$

Maple ✓
cpu = 0.078 (sec), leaf count = 68

$$\{-\frac{1}{2}\ln (\sqrt{y\left(x\right)}bx+a{x}^2-2y\left(x\right))+b\sqrt{y\left(x\right)}\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(1(b\sqrt{y\left(x\right)}+2ax)\frac{1}{\sqrt{y\left(x\right)(b^2+8a)}}\right)\frac{1}{\sqrt{y\left(x\right)(b^2+8a)}}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

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

Mathematica raw output

Solve[((2*b^3*ArcTanh[(b^2 - 4*a*Sqrt[(b^2*y[x])/(a^2*x^2)])/(b*Sqrt[8*a + b^2])
])/Sqrt[8*a + b^2] + 2*a*C[1] + 2*b^2*Log[x] + b^2*Log[b^2*(-1 + (2*y[x])/(a*x^2
) - Sqrt[(b^2*y[x])/(a^2*x^2)])])/a == 0, y[x]]

Maple raw input

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

Maple raw output

-1/2*ln(y(x)^(1/2)*b*x+a*x^2-2*y(x))+b*y(x)^(1/2)/(y(x)*(b^2+8*a))^(1/2)*arctanh
((b*y(x)^(1/2)+2*a*x)/(y(x)*(b^2+8*a))^(1/2))+_C1 = 0