##### 4.3.1 $$y'(x)=a x+b \sqrt {y(x)}$$

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

[[_homogeneous, class G], _Chini]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.519533 (sec), leaf count = 113

$\text {Solve}\left [\frac {b^2 \log \left (b^2 \left (\sqrt {\frac {b^2 y(x)}{a^2 x^2}}-\frac {2 y(x)}{a x^2}+1\right )\right )+\frac {2 b^3 \tanh ^{-1}\left (\frac {b \left (1-\frac {4 a \sqrt {\frac {b^2 y(x)}{a^2 x^2}}}{b^2}\right )}{\sqrt {8 a+b^2}}\right )}{\sqrt {8 a+b^2}}+2 a c_1+2 b^2 \log (x)}{a}=0,y(x)\right ]$

Maple
cpu = 0.072 (sec), leaf count = 68

$\left [-\frac {\ln \left (\sqrt {y \left (x \right )}\, b x +a \,x^{2}-2 y \left (x \right )\right )}{2}+\frac {b \sqrt {y \left (x \right )}\, \arctanh \left (\frac {b \sqrt {y \left (x \right )}+2 a x}{\sqrt {y \left (x \right ) \left (b^{2}+8 a \right )}}\right )}{\sqrt {y \left (x \right ) \left (b^{2}+8 a \right )}}+\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

Solve[((2*b^3*ArcTanh[(b*(1 - (4*a*Sqrt[(b^2*y[x])/(a^2*x^2)])/b^2))/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))

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)*arctan
h((b*y(x)^(1/2)+2*a*x)/(y(x)*(b^2+8*a))^(1/2))+_C1 = 0]