89.16.32 problem 32

Internal problem ID [24682]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Exercises at page 140
Problem number : 32
Date solved : Thursday, October 02, 2025 at 10:47:02 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+5 x&=8 \sin \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=4 \\ x^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 25

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+5*x(t) = 8*sin(t); 
ic:=[x(0) = 4, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \left (5 \cos \left (t \right )+9 \sin \left (t \right )\right ) {\mathrm e}^{-2 t}+\sin \left (t \right )-\cos \left (t \right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 30

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

\begin{align*} x(t)&\to \left (9 e^{-2 t}+1\right ) \sin (t)+\left (5 e^{-2 t}-1\right ) \cos (t) \end{align*}

✓ Sympy. Time used: 0.145 (sec). Leaf size: 24

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

\[ x{\left (t \right )} = \left (9 \sin {\left (t \right )} + 5 \cos {\left (t \right )}\right ) e^{- 2 t} + \sin {\left (t \right )} - \cos {\left (t \right )} \]