89.13.20 problem 20

Internal problem ID [24603]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 8. Linear Differential Equations with constant coefficients. Exercises at page 127
Problem number : 20
Date solved : Thursday, October 02, 2025 at 10:46:26 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} 4 y+4 y^{\prime }+y^{\prime \prime }+y^{\prime \prime \prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=-1 \\ y^{\prime \prime }\left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 15

ode:=diff(diff(diff(y(x),x),x),x)+diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = -1, (D@@2)(y)(0) = 5]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 17

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

\begin{align*} y(x)&\to e^{-x}-\cos (2 x) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): -1, Subs(Derivative(y(x), (x, 2)), x, 0): 5} 
dsolve(ode,func=y(x),ics=ics)
 

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