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 = 0.0345841 (sec), leaf count = 43

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

Maple
cpu = 0.018 (sec), leaf count = 40

\[ \left \{ y \relax (x ) = \left ({\frac {bx}{{a}^{2}}{\frac {1}{\sqrt {{a}^{2}+{x}^{2}}}}}+{\it \_C1} \right ) \left (x\sqrt {{a}^{2}+{x}^{2}}+{a}^{2}+{x}^{2} \right ) \right \} \] 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] -> ((x + Sqrt[a^2 + x^2])*(b*x + a^2*Sqrt[a^2 + x^2]*C[1]))/a^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),'implicit')

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)