54.5.5 problem 1538

Internal problem ID [12801]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1538
Date solved : Wednesday, October 01, 2025 at 02:22:09 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+2 a^{2} y^{\prime \prime }+a^{4} y-\cosh \left (a x \right )&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 42

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

\[ y = \frac {4 a^{4} \left (c_3 x +c_1 \right ) \cos \left (a x \right )+4 a^{4} \left (c_4 x +c_2 \right ) \sin \left (a x \right )+\cosh \left (a x \right )}{4 a^{4}} \]

✓ Mathematica. Time used: 0.19 (sec). Leaf size: 176

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

\begin{align*} y(x)&\to \cos (a x) \int _1^x\frac {\cosh (a K[1]) (a \cos (a K[1]) K[1]-\sin (a K[1]))}{2 a^3}dK[1]+\sin (a x) \int _1^x\frac {\cosh (a K[3]) (\cos (a K[3])+a K[3] \sin (a K[3]))}{2 a^3}dK[3]+x \cos (a x) \int _1^x-\frac {\cos (a K[2]) \cosh (a K[2])}{2 a^2}dK[2]+x \sin (a x) \int _1^x-\frac {\cosh (a K[4]) \sin (a K[4])}{2 a^2}dK[4]+c_1 \cos (a x)+c_2 x \cos (a x)+c_3 \sin (a x)+c_4 x \sin (a x) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = \left (C_{1} + C_{2} x\right ) e^{- i a x} + \left (C_{3} + C_{4} x\right ) e^{i a x} + \frac {\cosh {\left (a x \right )}}{4 a^{4}} \]