32.4.16 problem 17

Internal problem ID [7789]
Book : Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section : Program 25. Second order differential equations. Further problems 25. page 1094
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 05:05:28 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+2 x^{\prime }+2 x&=85 \sin \left (3 t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=-20 \\ \end{align*}

✓ Maple. Time used: 0.098 (sec). Leaf size: 31

ode:=diff(diff(x(t),t),t)+2*diff(x(t),t)+2*x(t) = 85*sin(3*t); 
ic:=[x(0) = 0, D(x)(0) = -20]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \left (7 \sin \left (t \right )+6 \cos \left (t \right )\right ) {\mathrm e}^{-t}-6 \cos \left (3 t \right )-7 \sin \left (3 t \right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 36

ode=D[x[t],{t,2}]+2*D[x[t],t]+2*x[t]==85*Sin[3*t]; 
ic={x[0]==0,Derivative[1][x][0 ]==-20}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 7 e^{-t} \sin (t)-7 \sin (3 t)+6 e^{-t} \cos (t)-6 \cos (3 t) \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(2*x(t) - 85*sin(3*t) + 2*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): -20} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (7 \sin {\left (t \right )} + 6 \cos {\left (t \right )}\right ) e^{- t} - 7 \sin {\left (3 t \right )} - 6 \cos {\left (3 t \right )} \]