ODE
\[ 4 \left (x^2+1\right ) y''(x)=x^2+4 x y'(x) \] ODE Classification
[[_2nd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0716702 (sec), leaf count = 57
\[\left \{\left \{y(x)\to \frac {1}{16} \left (8 c_1 \sqrt {x^2+1} x+8 c_1 \sinh ^{-1}(x)+16 c_2-3 x^2+2 \sqrt {x^2+1} x \sinh ^{-1}(x)+\sinh ^{-1}(x)^2\right )\right \}\right \}\]
Maple ✓
cpu = 0.117 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) ={\frac {x}{2} \left ( {\frac {{\it Arcsinh} \left ( x \right ) }{4}}+{\it \_C1} \right ) \sqrt {{x}^{2}+1}}-{\frac {3\,{x}^{2}}{16}}+{\frac {{\it \_C1}\,{\it Arcsinh} \left ( x \right ) }{2}}+{\frac { \left ( {\it Arcsinh} \left ( x \right ) \right ) ^{2}}{16}}+{\it \_C2}-{\frac {1}{16}} \right \} \] Mathematica raw input
DSolve[4*(1 + x^2)*y''[x] == x^2 + 4*x*y'[x],y[x],x]
Mathematica raw output
{{y[x] -> (-3*x^2 + 2*x*Sqrt[1 + x^2]*ArcSinh[x] + ArcSinh[x]^2 + 8*x*Sqrt[1 + x^2]*C[1] + 8*ArcSinh[x]*C[1] + 16*C[2])/16}}
Maple raw input
dsolve(4*(x^2+1)*diff(diff(y(x),x),x) = 4*x*diff(y(x),x)+x^2, y(x),'implicit')
Maple raw output
y(x) = 1/2*x*(1/4*arcsinh(x)+_C1)*(x^2+1)^(1/2)-3/16*x^2+1/2*_C1*arcsinh(x)+1/16*arcsinh(x)^2+_C2-1/16