60.5.6 problem 1539

Internal problem ID [11503]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1539
Date solved : Sunday, March 30, 2025 at 08:23:47 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+\left (\lambda +1\right ) a^{2} y^{\prime \prime }+\lambda \,a^{4} y&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 35

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+(lambda+1)*a^2*diff(diff(y(x),x),x)+lambda*a^4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sin \left (a x \right )+c_2 \cos \left (a x \right )+c_3 \sin \left (a \sqrt {\lambda }\, x \right )+c_4 \cos \left (a \sqrt {\lambda }\, x \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 44

ode=a^4*\[Lambda]*y[x] + a^2*(1 + \[Lambda])*D[y[x],{x,2}] + Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cos \left (a \sqrt {\lambda } x\right )+c_2 \sin \left (a \sqrt {\lambda } x\right )+c_3 \cos (a x)+c_4 \sin (a x) \]

✓ Sympy. Time used: 0.179 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(a**4*lambda_*y(x) + a**2*(lambda_ + 1)*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{- x \sqrt {- a^{2}}} + C_{2} e^{x \sqrt {- a^{2}}} + C_{3} e^{- x \sqrt {- a^{2} \lambda _{}}} + C_{4} e^{x \sqrt {- a^{2} \lambda _{}}} \]