60.6.1 problem 1578

Internal problem ID [11577]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 5, linear fifth and higher order
Problem number : 1578
Date solved : Tuesday, January 28, 2025 at 06:06:56 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} 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{align*}

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 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), singsol=all)
 

\[ y = {\mathrm e}^{a x} \left (\int {\mathrm e}^{-2 a x} \left (\int {\mathrm e}^{a x} \left (\operatorname {AiryAi}\left (-\frac {\left (\lambda \left (a x -b \right )+a^{2}\right ) \left (-a \lambda \right )^{{1}/{3}}}{a \lambda }\right ) c_3 +c_4 \operatorname {AiryBi}\left (-\frac {\left (\lambda \left (a x -b \right )+a^{2}\right ) \left (-a \lambda \right )^{{1}/{3}}}{a \lambda }\right )\right )d x +c_{2} \right )d x +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.151 (sec). Leaf size: 130

DSolve[a^4*y[x] - 2*a^2*D[y[x],{x,2}] - \[Lambda]*(-b + a*x)*(-(a^2*y[x]) + D[y[x],{x,2}]) + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-a x} \left (c_3 \int _1^x2 a e^{2 a K[1]} \int e^{-a K[1]} \operatorname {AiryAi}\left (\frac {a^2+\lambda K[1] a-b \lambda }{(a \lambda )^{2/3}}\right ) \, dK[1]dK[1]+c_4 \int _1^x2 a e^{2 a K[2]} \int e^{-a K[2]} \operatorname {AiryBi}\left (\frac {a^2+\lambda K[2] a-b \lambda }{(a \lambda )^{2/3}}\right ) \, dK[2]dK[2]+c_2 e^{2 a x}+c_1\right ) \]