ODE
\[ y'''(x)-2 \left (-2 a-4 x^2+1\right ) y'(x)-8 a x y(x)-6 x y''(x)=0 \] ODE Classification
[[_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.214547 (sec), leaf count = 57
\[\left \{\left \{y(x)\to c_2 H_{\frac {a}{2}}(x) \, _1F_1\left (-\frac {a}{4};\frac {1}{2};x^2\right )+c_1 H_{\frac {a}{2}}(x){}^2+c_3 \, _1F_1\left (-\frac {a}{4};\frac {1}{2};x^2\right ){}^2\right \}\right \}\]
Maple ✓
cpu = 0.634 (sec), leaf count = 64
\[\left [y \left (x \right ) = \textit {\_C1} \KummerM \left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2} x^{2}+\textit {\_C2} \KummerU \left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2} x^{2}+\textit {\_C3} \KummerM \left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) x^{2} \KummerU \left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )\right ]\] Mathematica raw input
DSolve[-8*a*x*y[x] - 2*(1 - 2*a - 4*x^2)*y'[x] - 6*x*y''[x] + y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*HermiteH[a/2, x]^2 + C[2]*HermiteH[a/2, x]*Hypergeometric1F1[-1/4*a, 1/2, x^2] + C[3]*Hypergeometric1F1[-1/4*a, 1/2, x^2]^2}}
Maple raw input
dsolve(diff(diff(diff(y(x),x),x),x)-6*x*diff(diff(y(x),x),x)-2*(-4*x^2-2*a+1)*diff(y(x),x)-8*a*x*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*KummerM(1/2-1/4*a,3/2,x^2)^2*x^2+_C2*KummerU(1/2-1/4*a,3/2,x^2)^2*x^2+_C3*KummerM(1/2-1/4*a,3/2,x^2)*x^2*KummerU(1/2-1/4*a,3/2,x^2)]