2.12.11 problem 12

Internal problem ID [918]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.6, Forced Oscillations and Resonance. Page 362
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 04:19:34 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+6 x^{\prime }+13 x&=10 \sin \left (5 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 36

ode:=diff(diff(x(t),t),t)+6*diff(x(t),t)+13*x(t) = 10*sin(5*t); 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {25 \left (2 \cos \left (2 t \right )+5 \sin \left (2 t \right )\right ) {\mathrm e}^{-3 t}}{174}-\frac {25 \cos \left (5 t \right )}{87}-\frac {10 \sin \left (5 t \right )}{87} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 49

ode=D[x[t],{t,2}]+6*D[x[t],t]+13*x[t]==10*Sin[5*t]; 
ic={x[0]==0,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {5}{174} e^{-3 t} \left (25 \sin (2 t)-4 e^{3 t} \sin (5 t)+10 \cos (2 t)-10 e^{3 t} \cos (5 t)\right ) \end{align*}

✓ Sympy. Time used: 0.171 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(13*x(t) - 10*sin(5*t) + 6*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {125 \sin {\left (2 t \right )}}{174} + \frac {25 \cos {\left (2 t \right )}}{87}\right ) e^{- 3 t} - \frac {10 \sin {\left (5 t \right )}}{87} - \frac {25 \cos {\left (5 t \right )}}{87} \]