\[ \boxed { {\it d4y} \left ( x \right ) -2\,{a}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +{a}^{4}y \left ( x \right ) -\lambda \, \left ( ax-b \right ) \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -{a}^{2}y \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 438.074128 (sec), leaf count = 139 \[ \left \{\left \{y(x)\to c_3 e^{-a x} \int _1^x 2 a e^{2 a K[1]} \int e^{-a K[1]} \text {Ai}\left (\frac {a^2+\lambda K[1] a-b \lambda }{(a \lambda )^{2/3}}\right ) \, dK[1] \, dK[1]+c_4 e^{-a x} \int _1^x 2 a e^{2 a K[2]} \int e^{-a K[2]} \text {Bi}\left (\frac {a^2+\lambda K[2] a-b \lambda }{(a \lambda )^{2/3}}\right ) \, dK[2] \, dK[2]+c_1 e^{-a x}+c_2 e^{a x}\right \}\right \} \]
Maple: cpu = 0.764 (sec), leaf count = 92 \[ \left \{ y \left ( x \right ) ={{\rm e}^{ax}} \left ( \int \!{{\rm e}^{-2 \,ax}} \left ( \int \!{\it \_C3}\,{{\rm e}^{ax}}{{\rm Ai}\left (-{\frac {\lambda \, \left ( ax-b \right ) +{a}^{2}}{a\lambda }\sqrt [3]{-a\lambda } }\right )}+{\it \_C4}\,{{\rm e}^{ax}}{{\rm Bi}\left (-{\frac {\lambda \, \left ( ax-b \right ) +{a}^{2}}{a\lambda }\sqrt [3]{-a\lambda }}\right )} \,{\rm d}x+{\it \_C2} \right ) \,{\rm d}x+{\it \_C1} \right ) \right \} \]