\[ a^4 y(x)-\lambda (a x-b) \left (y''(x)-a^2 y(x)\right )-2 a^2 y''(x)+y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 25.221 (sec), leaf count = 141
DSolve[a^4*y[x] - 2*a^2*Derivative[2][y][x] - lambda*(-b + a*x)*(-(a^2*y[x]) + Derivative[2][y][x]) + Derivative[4][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_3 e^{-a x} \int _1^x2 a e^{2 a K[1]} \int e^{-a K[1]} \operatorname {AiryAi}\left (\frac {a^2+\lambda K[1] a-b \lambda }{(a \lambda )^{2/3}}\right ) \, dK[1]dK[1]+c_4 e^{-a x} \int _1^x2 a e^{2 a K[2]} \int e^{-a K[2]} \operatorname {AiryBi}\left (\frac {a^2+\lambda K[2] a-b \lambda }{(a \lambda )^{2/3}}\right ) \, dK[2]dK[2]+c_1 e^{-a x}+c_2 e^{a x}\right \}\right \}\] ✓ Maple : cpu = 0.425 (sec), leaf count = 89
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)-2*a^2*diff(diff(y(x),x),x)+a^4*y(x)-lambda*(a*x-b)*(diff(diff(y(x),x),x)-a^2*y(x))=0,y(x))
\[y \left (x \right ) = {\mathrm e}^{a x} \left (\int {\mathrm e}^{-2 a x} \left (\int {\mathrm e}^{a x} \left (c_{4} \operatorname {AiryBi}\left (-\frac {\left (-a \lambda \right )^{\frac {1}{3}} \left (\lambda \left (a x -b \right )+a^{2}\right )}{\lambda a}\right )+c_{3} \operatorname {AiryAi}\left (-\frac {\left (-a \lambda \right )^{\frac {1}{3}} \left (\lambda \left (a x -b \right )+a^{2}\right )}{\lambda a}\right )\right )d x +c_{2}\right )d x +c_{1}\right )\]