ODE
\[ x^3 y'''(x)-\left (12-a x^3\right ) y(x)+6 x^2 y''(x)=0 \] ODE Classification
[[_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.66616 (sec), leaf count = 97
\[\left \{\left \{y(x)\to \frac {c_1 e^{-\sqrt [3]{a} x} \left (\sqrt [3]{a} x+2\right )+c_2 e^{\sqrt [3]{-1} \sqrt [3]{a} x} \left (\sqrt [3]{a} x+2 (-1)^{2/3}\right )+c_3 e^{-(-1)^{2/3} \sqrt [3]{a} x} \left (\sqrt [3]{a} x-2 \sqrt [3]{-1}\right )}{x^3}\right \}\right \}\]
Maple ✓
cpu = 0.371 (sec), leaf count = 134
\[ \left \{ y \left ( x \right ) ={\frac {1}{{x}^{3}} \left ( -{\it \_C2}\, \left ( \left ( -i-\sqrt {3} \right ) \left ( -{a}^{4} \right ) ^{{\frac {2}{3}}}+i{a}^{3}x \right ) {{\rm e}^{{\frac {{\frac {i}{2}} \left ( -\sqrt {3}+i \right ) x}{a}\sqrt [3]{-{a}^{4}}}}}+{\it \_C3}\, \left ( \left ( -i+\sqrt {3} \right ) \left ( -{a}^{4} \right ) ^{{\frac {2}{3}}}+i{a}^{3}x \right ) {{\rm e}^{{\frac {{\frac {i}{2}} \left ( \sqrt {3}+i \right ) x}{a}\sqrt [3]{-{a}^{4}}}}}+{\it \_C1}\,{{\rm e}^{{\frac {x}{a}\sqrt [3]{-{a}^{4}}}}} \left ( {a}^{3}x+2\, \left ( -{a}^{4} \right ) ^{2/3} \right ) \right ) } \right \} \] Mathematica raw input
DSolve[-((12 - a*x^3)*y[x]) + 6*x^2*y''[x] + x^3*y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (((2 + a^(1/3)*x)*C[1])/E^(a^(1/3)*x) + E^((-1)^(1/3)*a^(1/3)*x)*(2*(-1)^(2/3) + a^(1/3)*x)*C[2] + ((-2*(-1)^(1/3) + a^(1/3)*x)*C[3])/E^((-1)^(2/3)*a^(1/3)*x))/x^3}}
Maple raw input
dsolve(x^3*diff(diff(diff(y(x),x),x),x)+6*x^2*diff(diff(y(x),x),x)-(-a*x^3+12)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (-_C2*((-I-3^(1/2))*(-a^4)^(2/3)+I*a^3*x)*exp(1/2*I*(-3^(1/2)+I)*(-a^4)^(1/3)*x/a)+_C3*((-I+3^(1/2))*(-a^4)^(2/3)+I*a^3*x)*exp(1/2*I*(3^(1/2)+I)*(-a^4)^(1/3)*x/a)+_C1*exp((-a^4)^(1/3)/a*x)*(a^3*x+2*(-a^4)^(2/3)))/x^3