12.18.26 problem section 9.2, problem 26

Internal problem ID [2140]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.2. constant coefficient. Page 483
Problem number : section 9.2, problem 26
Date solved : Saturday, March 29, 2025 at 11:49:21 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-2 y^{\prime \prime }-8 y^{\prime }-8 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=5\\ y^{\prime }\left (0\right )&=-2\\ y^{\prime \prime }\left (0\right )&=6\\ y^{\prime \prime \prime }\left (0\right )&=8 \end{align*}

✓ Maple. Time used: 0.085 (sec). Leaf size: 25

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+2*diff(diff(diff(y(x),x),x),x)-2*diff(diff(y(x),x),x)-8*diff(y(x),x)-8*y(x) = 0; 
ic:=y(0) = 5, D(y)(0) = -2, (D@@2)(y)(0) = 6, (D@@3)(y)(0) = 8; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 31

ode=D[y[x],{x,4}]+2*D[y[x],{x,3}]-2*D[y[x],{x,2}]-8*D[y[x],x]-8*y[x]==0; 
ic={y[0]==5,Derivative[1][y][0] ==-2,Derivative[2][y][0] ==6,Derivative[3][y][0]==8}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.312 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*y(x) - 8*Derivative(y(x), x) - 2*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 5, Subs(Derivative(y(x), x), x, 0): -2, Subs(Derivative(y(x), (x, 2)), x, 0): 6, Subs(Derivative(y(x), (x, 3)), x, 0): 8} 
dsolve(ode,func=y(x),ics=ics)
 

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