##### 4.7.9 $$\left (a^2+x^2\right ) y'(x)=\left (\sqrt {a^2+x^2}+x\right ) (b+y(x))$$

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.413936 (sec), leaf count = 60

$\left \{\left \{y(x)\to \frac {\left (x \left (\sqrt {a^2+x^2}+x\right )+a^2\right ) \left (b x+a^2 c_1 \sqrt {a^2+x^2}\right )}{a^2 \sqrt {a^2+x^2}}\right \}\right \}$

Maple
cpu = 0.021 (sec), leaf count = 40

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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