6.19.57 problem section 9.3, problem 57

Internal problem ID [2204]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined Coefficients for Higher Order Equations. Page 495
Problem number : section 9.3, problem 57
Date solved : Tuesday, September 30, 2025 at 05:24:58 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y&=\sinh \left (x \right ) \cos \left (x \right )-\cosh \left (x \right ) \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 50

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+4*y(x) = sinh(x)*cos(x)-cosh(x)*sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_3 \cos \left (x \right )+c_4 \sin \left (x \right )\right ) {\mathrm e}^{-x}+\frac {\left (4 x \sinh \left (x \right )+32 c_2 \,{\mathrm e}^{x}-3 \cosh \left (x \right )\right ) \sin \left (x \right )}{32}+\frac {3 \left (\frac {32 c_1 \,{\mathrm e}^{x}}{3}+\sinh \left (x \right )\right ) \cos \left (x \right )}{32} \]

✓ Mathematica. Time used: 0.356 (sec). Leaf size: 63

ode=D[y[x],{x,4}]+0*D[y[x],{x,3}]-0*D[y[x],{x,2}]-0*D[y[x],x]+4*y[x]==Sinh[x]*Cos[x]-Cosh[x]*Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{64} e^{-x} \left (\left ((3+64 c_4) e^{2 x}-3+64 c_1\right ) \cos (x)+\left (-4 x+e^{2 x} (4 x-3+64 c_3)-3+64 c_2\right ) \sin (x)\right ) \end{align*}

✓ Sympy. Time used: 0.152 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = \frac {x \sin {\left (x \right )} \sinh {\left (x \right )}}{8} + \left (C_{1} \sin {\left (x \right )} + C_{2} \cos {\left (x \right )}\right ) e^{- x} + \left (C_{3} \sin {\left (x \right )} + C_{4} \cos {\left (x \right )}\right ) e^{x} \]