##### 4.8.45 $$\sqrt {a^2+x^2} y'(x)+y(x)+x=\sqrt {a^2+x^2}$$

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

[_linear]

Book solution method
Linear ODE

Mathematica
cpu = 0.995474 (sec), leaf count = 128

$\left \{\left \{y(x)\to \frac {\sqrt {1-\frac {x}{\sqrt {a^2+x^2}}} \left (\int _1^x\frac {\sqrt {\frac {K[1]}{\sqrt {a^2+K[1]^2}}+1} \left (\sqrt {a^2+K[1]^2}-K[1]\right )}{\sqrt {a^2+K[1]^2} \sqrt {1-\frac {K[1]}{\sqrt {a^2+K[1]^2}}}}dK[1]+c_1\right )}{\sqrt {\frac {x}{\sqrt {a^2+x^2}}+1}}\right \}\right \}$

Maple
cpu = 0.016 (sec), leaf count = 36

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

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

Mathematica raw output

{{y[x] -> (Sqrt[1 - x/Sqrt[a^2 + x^2]]*(C[1] + Inactive[Integrate][(Sqrt[1 + K[1
]/Sqrt[a^2 + K[1]^2]]*(-K[1] + Sqrt[a^2 + K[1]^2]))/(Sqrt[a^2 + K[1]^2]*Sqrt[1 -
 K[1]/Sqrt[a^2 + K[1]^2]]), {K[1], 1, x}]))/Sqrt[1 + x/Sqrt[a^2 + x^2]]}}

Maple raw input

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

Maple raw output

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