12.18.24 problem section 9.2, problem 24

Internal problem ID [2138]
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 24
Date solved : Saturday, March 29, 2025 at 11:49:19 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+7 y^{\prime \prime }+6 y^{\prime }-8 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-2\\ y^{\prime }\left (0\right )&=-8\\ y^{\prime \prime }\left (0\right )&=-14\\ y^{\prime \prime \prime }\left (0\right )&=-62 \end{align*}

✓ Maple. Time used: 0.059 (sec). Leaf size: 25

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-6*diff(diff(diff(y(x),x),x),x)+7*diff(diff(y(x),x),x)+6*diff(y(x),x)-8*y(x) = 0; 
ic:=y(0) = -2, D(y)(0) = -8, (D@@2)(y)(0) = -14, (D@@3)(y)(0) = -62; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = {\mathrm e}^{2 x}+2 \,{\mathrm e}^{-x}-4 \,{\mathrm e}^{x}-{\mathrm e}^{4 x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 30

ode=D[y[x],{x,4}]-6*D[y[x],{x,3}]+7*D[y[x],{x,2}]+6*D[y[x],x]-8*y[x]==0; 
ic={y[0]==-2,Derivative[1][y][0] ==-8,Derivative[2][y][0] ==-14,Derivative[3][y][0]==-62}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to 2 e^{-x}-4 e^x+e^{2 x}-e^{4 x} \]

✓ Sympy. Time used: 0.225 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*y(x) + 6*Derivative(y(x), x) + 7*Derivative(y(x), (x, 2)) - 6*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): -2, Subs(Derivative(y(x), x), x, 0): -8, Subs(Derivative(y(x), (x, 2)), x, 0): -14, Subs(Derivative(y(x), (x, 3)), x, 0): -62} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - e^{4 x} + e^{2 x} - 4 e^{x} + 2 e^{- x} \]