Internal problem ID [9157]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 5, linear fifth and higher order
Problem number: 1578.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+a^{4} y-\lambda \left (a x -b \right ) \left (y^{\prime \prime }-a^{2} y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 92
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), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{a x} \left (\int {\mathrm e}^{-2 a x} \left (\int \left (c_{3} {\mathrm e}^{a x} \AiryAi \left (-\frac {\left (\lambda \left (a x -b \right )+a^{2}\right ) \left (-a \lambda \right )^{\frac {1}{3}}}{a \lambda }\right )+c_{4} {\mathrm e}^{a x} \AiryBi \left (-\frac {\left (\lambda \left (a x -b \right )+a^{2}\right ) \left (-a \lambda \right )^{\frac {1}{3}}}{a \lambda }\right )\right )d x +c_{2}\right )d x +c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 27.152 (sec). Leaf size: 130
DSolve[a^4*y[x] - 2*a^2*y''[x] - \[Lambda]*(-b + a*x)*(-(a^2*y[x]) + y''[x]) + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-a x} \left (c_3 \int _1^x2 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 \int _1^x2 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_2 e^{2 a x}+c_1\right ) \\ \end{align*}