6.19.45 problem section 9.3, problem 45

Internal problem ID [2192]
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 45
Date solved : Tuesday, September 30, 2025 at 05:24:49 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+8 y^{\prime \prime }+8 y^{\prime }+4 y&=-2 \,{\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right ) \end{align*}

✓ Maple. Time used: 0.075 (sec). Leaf size: 36

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+4*diff(diff(diff(y(x),x),x),x)+8*diff(diff(y(x),x),x)+8*diff(y(x),x)+4*y(x) = -2*exp(x)*(cos(x)-sin(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\left (c_3 x +c_1 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_4 x +c_2 \right )\right ) {\mathrm e}^{-x}-\frac {{\mathrm e}^{x} \left (\sin \left (x \right )+\cos \left (x \right )\right )}{16} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 54

ode=1*D[y[x],{x,4}]+4*D[y[x],{x,3}]+8*D[y[x],{x,2}]+8*D[y[x],x]+4*y[x]==-2*Exp[x]*(Cos[1*x]-Sin[1*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{16} e^{-x} \left (\left (-e^{2 x}+16 (c_4 x+c_3)\right ) \cos (x)-\left (e^{2 x}-16 (c_2 x+c_1)\right ) \sin (x)\right ) \end{align*}

✓ Sympy. Time used: 0.274 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-2*sin(x) + 2*cos(x))*exp(x) + 4*y(x) + 8*Derivative(y(x), x) + 8*Derivative(y(x), (x, 2)) + 4*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\left (C_{1} + C_{2} x\right ) \sin {\left (x \right )} + \left (C_{3} + C_{4} x\right ) \cos {\left (x \right )}\right ) e^{- x} - \frac {\sqrt {2} e^{x} \sin {\left (x + \frac {\pi }{4} \right )}}{16} \]