ODE
\[ x \left (x^2+2\right ) y''(x)-y'(x)-6 x y(x)=0 \] ODE Classification
[[_2nd_order, _exact, _linear, _homogeneous]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.206294 (sec), leaf count = 100
\[\left \{\left \{y(x)\to \frac {-\sqrt [4]{2} c_2 \left (x^2+2\right ) x^2 \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {x^2}{2}\right )+3 c_1 x^{7/2}+6 c_1 x^{3/2}-c_2 \sqrt [4]{x^2+2} x^2-c_2 \sqrt [4]{x^2+2}}{3 \sqrt [4]{x^2+2}}\right \}\right \}\]
Maple ✓
cpu = 0.123 (sec), leaf count = 31
\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}+2 \right ) ^{{\frac {3}{4}}} \left ( {x}^{{\frac {3}{2}}}{\it \_C1}+{\mbox {$_2$F$_1$}(-{\frac {3}{4}},{\frac {7}{4}};\,{\frac {1}{4}};\,-{\frac {{x}^{2}}{2}})}{\it \_C2} \right ) \right \} \] Mathematica raw input
DSolve[-6*x*y[x] - y'[x] + x*(2 + x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (6*x^(3/2)*C[1] + 3*x^(7/2)*C[1] - (2 + x^2)^(1/4)*C[2] - x^2*(2 + x^2)^(1/4)*C[2] - 2^(1/4)*x^2*(2 + x^2)*C[2]*Hypergeometric2F1[1/4, 3/4, 5/4, -x^2/2])/(3*(2 + x^2)^(1/4))}}
Maple raw input
dsolve(x*(x^2+2)*diff(diff(y(x),x),x)-diff(y(x),x)-6*x*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (x^2+2)^(3/4)*(x^(3/2)*_C1+hypergeom([-3/4, 7/4],[1/4],-1/2*x^2)*_C2)