ODE
\[ x \left (x^2+1\right ) y'''(x)+3 \left (2 x^2+1\right ) y''(x)-12 y(x)=0 \] ODE Classification
[[_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.252613 (sec), leaf count = 96
\[\left \{\left \{y(x)\to \frac {4 c_1 x^3+2 c_2 \sqrt {x^2+1} x^2+3 c_3 x^2+3 c_3 \sqrt {x^2+1} x^2 \log (x)-3 c_3 \sqrt {x^2+1} x^2 \log \left (\sqrt {x^2+1}+1\right )+2 c_1 x+c_3}{6 x}\right \}\right \}\]
Maple ✓
cpu = 0.251 (sec), leaf count = 60
\[ \left \{ y \left ( x \right ) ={\frac {1}{x} \left ( 3\,{\it Artanh} \left ( {\frac {1}{\sqrt {{x}^{2}+1}}} \right ) \sqrt {{x}^{2}+1}{\it \_C2}\,{x}^{2}+{\it \_C1}\,{x}^{2}\sqrt {{x}^{2}+1}+2\,{\it \_C3}\,{x}^{3}-3\,{\it \_C2}\,{x}^{2}+{\it \_C3}\,x-{\it \_C2} \right ) } \right \} \] Mathematica raw input
DSolve[-12*y[x] + 3*(1 + 2*x^2)*y''[x] + x*(1 + x^2)*y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2*x*C[1] + 4*x^3*C[1] + 2*x^2*Sqrt[1 + x^2]*C[2] + C[3] + 3*x^2*C[3] + 3*x^2*Sqrt[1 + x^2]*C[3]*Log[x] - 3*x^2*Sqrt[1 + x^2]*C[3]*Log[1 + Sqrt[1 + x^2]])/(6*x)}}
Maple raw input
dsolve(x*(x^2+1)*diff(diff(diff(y(x),x),x),x)+3*(2*x^2+1)*diff(diff(y(x),x),x)-12*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (3*arctanh(1/(x^2+1)^(1/2))*(x^2+1)^(1/2)*_C2*x^2+_C1*x^2*(x^2+1)^(1/2)+2*_C3*x^3-3*_C2*x^2+_C3*x-_C2)/x