ODE
\[ x^2 y'''(x)+a x^2 y(x)-6 y'(x)=0 \] ODE Classification
[[_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.628746 (sec), leaf count = 97
\[\left \{\left \{y(x)\to \frac {c_1 e^{-\sqrt [3]{a} x} \left (\sqrt [3]{a} x+2\right )+c_2 e^{\sqrt [3]{-1} \sqrt [3]{a} x} \left (\sqrt [3]{a} x+2 (-1)^{2/3}\right )+c_3 e^{-(-1)^{2/3} \sqrt [3]{a} x} \left (\sqrt [3]{a} x-2 \sqrt [3]{-1}\right )}{x}\right \}\right \}\]
Maple ✓
cpu = 1.303 (sec), leaf count = 151
\[\left [y \left (x \right ) = \frac {\textit {\_C1} \left (a^{3} x +2 \left (-a^{4}\right )^{\frac {2}{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} x}{a}}}{x}+\frac {\textit {\_C2} \,{\mathrm e}^{\frac {i \left (i-\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {1}{3}} x}{2 a}} \left (\left (-a^{4}\right )^{\frac {2}{3}} \sqrt {3}-i a^{3} x +i \left (-a^{4}\right )^{\frac {2}{3}}\right )}{x}+\frac {\textit {\_C3} \,{\mathrm e}^{\frac {i \left (\sqrt {3}+i\right ) \left (-a^{4}\right )^{\frac {1}{3}} x}{2 a}} \left (-i a^{3} x +i \left (-a^{4}\right )^{\frac {2}{3}}-\left (-a^{4}\right )^{\frac {2}{3}} \sqrt {3}\right )}{x}\right ]\] Mathematica raw input
DSolve[a*x^2*y[x] - 6*y'[x] + x^2*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}}
Maple raw input
dsolve(x^2*diff(diff(diff(y(x),x),x),x)-6*diff(y(x),x)+a*x^2*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1/x*(a^3*x+2*(-a^4)^(2/3))*exp((-a^4)^(1/3)/a*x)+_C2*exp(1/2*I*(I-3^(1/2))*(-a^4)^(1/3)*x/a)*((-a^4)^(2/3)*3^(1/2)-I*a^3*x+I*(-a^4)^(2/3))/x+_C3*exp(1/2*I*(3^(1/2)+I)*(-a^4)^(1/3)*x/a)*(-I*a^3*x+I*(-a^4)^(2/3)-(-a^4)^(2/3)*3^(1/2))/x]