67.16.21 problem 24.4 (c)

Internal problem ID [16820]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 24. Variation of parameters. Additional exercises page 444
Problem number : 24.4 (c)
Date solved : Thursday, October 02, 2025 at 01:39:24 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-81 y&=\sinh \left (x \right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 33

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-81*y(x) = sinh(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\sinh \left (x \right )}{80}+c_1 \cos \left (3 x \right )+c_2 \,{\mathrm e}^{-3 x}+c_3 \,{\mathrm e}^{3 x}+c_4 \sin \left (3 x \right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 100

ode=D[y[x],{x,4}]-81*y[x]==Sinh[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \cos (3 x) \int _1^x\frac {1}{54} \sin (3 K[1]) \sinh (K[1])dK[1]+\sin (3 x) \int _1^x-\frac {1}{54} \cos (3 K[2]) \sinh (K[2])dK[2]+\frac {e^{-x}}{288}-\frac {e^x}{288}+c_1 e^{3 x}+c_3 e^{-3 x}+c_2 \cos (3 x)+c_4 \sin (3 x) \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-81*y(x) - sinh(x) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{- 3 x} + C_{2} e^{3 x} + C_{3} \sin {\left (3 x \right )} + C_{4} \cos {\left (3 x \right )} - \frac {\sinh {\left (x \right )}}{80} \]