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 = 0.221863 (sec), leaf count = 116

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

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

\[ \left \{ -{\frac {1}{2}\ln \left (\sqrt {y \relax (x ) }bx+a{x}^{2}-2\,y \relax (x ) \right ) }+{b\sqrt {y \relax (x ) }{\it Artanh} \left ({1 \left (b\sqrt {y \relax (x ) }+2\,ax \right ) {\frac {1}{\sqrt {y \relax (x ) \left ({b}^{2}+8\,a \right ) }}}} \right ) {\frac {1}{\sqrt {y \relax (x ) \left ({b}^{2}+8\,a \right ) }}}}+{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

Solve[((2*b^3*ArcTanh[(b^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