4.5.1 $$x y'(x)=a \sqrt {b^2 x^2+y(x)^2}+y(x)$$

ODE
$x y'(x)=a \sqrt {b^2 x^2+y(x)^2}+y(x)$ ODE Classiﬁcation

[[_homogeneous, class A], _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.39042 (sec), leaf count = 73

$\left \{\left \{y(x)\to \frac {1}{2} b e^{-c_1} \left (x^{1-a}-e^{2 c_1} x^{a+1}\right )\right \},\left \{y(x)\to \frac {1}{2} b e^{-c_1} x^{1-a} \left (-1+e^{2 c_1} x^{2 a}\right )\right \}\right \}$

Maple
cpu = 0.059 (sec), leaf count = 41

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

DSolve[x*y'[x] == y[x] + a*Sqrt[b^2*x^2 + y[x]^2],y[x],x]

Mathematica raw output

{{y[x] -> (b*(x^(1 - a) - E^(2*C[1])*x^(1 + a)))/(2*E^C[1])}, {y[x] -> (b*x^(1 -
 a)*(-1 + E^(2*C[1])*x^(2*a)))/(2*E^C[1])}}

Maple raw input

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

Maple raw output

[1/(x^a)/x*y(x)+1/(x^a)/x*(y(x)^2+b^2*x^2)^(1/2)-_C1 = 0]