ODE
\[ a+\left (x^2+1\right ) y''(x)-x y'(x)=0 \] ODE Classification
[[_2nd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0477726 (sec), leaf count = 38
\[\left \{\left \{y(x)\to \frac {1}{2} \left (-a x^2+c_1 \sqrt {x^2+1} x+c_1 \sinh ^{-1}(x)+2 c_2\right )\right \}\right \}\]
Maple ✓
cpu = 0.029 (sec), leaf count = 28
\[ \left \{ y \left ( x \right ) ={\frac {{\it \_C1}\,x}{2}\sqrt {{x}^{2}+1}}+{\frac {{\it \_C1}\,{\it Arcsinh} \left ( x \right ) }{2}}-{\frac {a{x}^{2}}{2}}+{\it \_C2} \right \} \] Mathematica raw input
DSolve[a - x*y'[x] + (1 + x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-(a*x^2) + x*Sqrt[1 + x^2]*C[1] + ArcSinh[x]*C[1] + 2*C[2])/2}}
Maple raw input
dsolve((x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+a = 0, y(x),'implicit')
Maple raw output
y(x) = 1/2*_C1*x*(x^2+1)^(1/2)+1/2*_C1*arcsinh(x)-1/2*a*x^2+_C2