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 = 0.101859 (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 = 0.23 (sec), leaf count = 59
\[ \left \{ y \left ( x \right ) ={x}^{2} \left ( \left ( {{\sl U}\left ({\frac {1}{2}}-{\frac {a}{4}},\,{\frac {3}{2}},\,{x}^{2}\right )} \right ) ^{2}{\it \_C2}+{{\sl U}\left ({\frac {1}{2}}-{\frac {a}{4}},\,{\frac {3}{2}},\,{x}^{2}\right )}{{\sl M}\left ({\frac {1}{2}}-{\frac {a}{4}},\,{\frac {3}{2}},\,{x}^{2}\right )}{\it \_C3}+ \left ( {{\sl M}\left ({\frac {1}{2}}-{\frac {a}{4}},\,{\frac {3}{2}},\,{x}^{2}\right )} \right ) ^{2}{\it \_C1} \right ) \right \} \] 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[-a/4, 1/2, x^2] + C[3]*Hypergeometric1F1[-a/4, 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),'implicit')
Maple raw output
y(x) = x^2*(KummerU(1/2-1/4*a,3/2,x^2)^2*_C2+KummerU(1/2-1/4*a,3/2,x^2)*KummerM(1/2-1/4*a,3/2,x^2)*_C3+KummerM(1/2-1/4*a,3/2,x^2)^2*_C1)