60.6.1 problem 1578

Internal problem ID [11538]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 5, linear fifth and higher order
Problem number : 1578
Date solved : Sunday, March 30, 2025 at 08:24:26 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*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 89

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{a x} \left (\int {\mathrm e}^{-2 a x} \left (\int {\mathrm e}^{a x} \left (c_3 \operatorname {AiryAi}\left (-\frac {\left (-a \lambda \right )^{{1}/{3}} \left (\lambda \left (a x -b \right )+a^{2}\right )}{a \lambda }\right )+c_4 \operatorname {AiryBi}\left (-\frac {\left (-a \lambda \right )^{{1}/{3}} \left (\lambda \left (a x -b \right )+a^{2}\right )}{a \lambda }\right )\right )d x +c_2 \right )d x +c_1 \right ) \]

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

ode=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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(a**4*y(x) - 2*a**2*Derivative(y(x), (x, 2)) - lambda_*(a*x - b)*(-a**2*y(x) + Derivative(y(x), (x, 2))) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve a**4*y(x) - 2*a**2*Derivative(y(x), (x, 2)) - lambda_*(a*x - b)*(-a**2*y(x) + Derivative(y(x), (x, 2))) + Derivative(y(x), (x, 4))