12.18.27 problem section 9.2, problem 27

Internal problem ID [2141]
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 27
Date solved : Tuesday, March 04, 2025 at 01:50:45 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} 4 y^{\prime \prime \prime \prime }+8 y^{\prime \prime \prime }+19 y^{\prime \prime }+32 y^{\prime }+12 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=-3\\ y^{\prime \prime }\left (0\right )&=-{\frac {7}{2}}\\ y^{\prime \prime \prime }\left (0\right )&={\frac {31}{4}} \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 21

ode:=4*diff(diff(diff(diff(y(x),x),x),x),x)+8*diff(diff(diff(y(x),x),x),x)+19*diff(diff(y(x),x),x)+32*diff(y(x),x)+12*y(x) = 0; 
ic:=y(0) = 3, D(y)(0) = -3, (D@@2)(y)(0) = -7/2, (D@@3)(y)(0) = 31/4; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ 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: 0.260 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(12*y(x) + 32*Derivative(y(x), x) + 19*Derivative(y(x), (x, 2)) + 8*Derivative(y(x), (x, 3)) + 4*Derivative(y(x), (x, 4)),0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): -3, Subs(Derivative(y(x), (x, 2)), x, 0): -7/2, Subs(Derivative(y(x), (x, 3)), x, 0): 31/4} 
dsolve(ode,func=y(x),ics=ics)
 

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