28.2.28 problem 28

Internal problem ID [4471]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 28
Date solved : Sunday, March 30, 2025 at 03:23:26 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+16 y&=64 \cos \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 69

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+16*y(x) = 64*cos(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+c_1 \,{\mathrm e}^{\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )+c_4 \,{\mathrm e}^{-\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+c_3 \,{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )+2 \cos \left (2 x \right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 82

ode=D[y[x],{x,4}]+16*y[x]==64*Cos[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-\sqrt {2} x} \left (2 e^{\sqrt {2} x} \cos (2 x)+\left (c_1 e^{2 \sqrt {2} x}+c_2\right ) \cos \left (\sqrt {2} x\right )+\left (c_4 e^{2 \sqrt {2} x}+c_3\right ) \sin \left (\sqrt {2} x\right )\right ) \]

✓ Sympy. Time used: 0.177 (sec). Leaf size: 66

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(16*y(x) - 64*cos(2*x) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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