67.12.16 problem 19.3 (d)

Internal problem ID [16655]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 19. Arbitrary Homogeneous linear equations with constant coefficients. Additional exercises page 369
Problem number : 19.3 (d)
Date solved : Thursday, October 02, 2025 at 01:37:23 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }+9 y^{\prime \prime }+9 y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=10 \\ y^{\prime }\left (0\right )&=0 \\ y^{\prime \prime }\left (0\right )&=6 \\ y^{\prime \prime \prime }\left (0\right )&=-60 \\ \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 18

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+diff(diff(diff(y(x),x),x),x)+9*diff(diff(y(x),x),x)+9*diff(y(x),x) = 0; 
ic:=[y(0) = 10, D(y)(0) = 0, (D@@2)(y)(0) = 6, (D@@3)(y)(0) = -60]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 4+6 \,{\mathrm e}^{-x}+2 \sin \left (3 x \right ) \]

✓ Mathematica. Time used: 0.097 (sec). Leaf size: 55

ode=D[y[x],{x,4}]+D[y[x],{x,3}]+9*D[y[x],{x,2}]+9*D[y[x],x]==0; 
ic={y[0]==10,Derivative[1][y][0] ==0,Derivative[2][y][0] ==6,Derivative[3][y][0]==-60}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\left (6 \cos (3 K[1])-6 e^{-K[1]}\right )dK[1]-\int _1^0\left (6 \cos (3 K[1])-6 e^{-K[1]}\right )dK[1]+10 \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*Derivative(y(x), x) + 9*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 10, Subs(Derivative(y(x), x), x, 0): 0, Subs(Derivative(y(x), (x, 2)), x, 0): 6, Subs(Derivative(y(x), (x, 3)), x, 0): -60} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 2 \sin {\left (3 x \right )} + 4 + 6 e^{- x} \]