78.4.9 problem 12.a

Internal problem ID [21037]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 5, Laplace transforms. Problems section 5.7
Problem number : 12.a
Date solved : Thursday, October 02, 2025 at 07:01:40 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.482 (sec). Leaf size: 73

ode:=[diff(x(t),t) = 2*x(t)+3*y(t)+2*sin(2*t), diff(y(t),t) = -3*x(t)+2*y(t)]; 
ic:=[x(0) = 1, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {141 \,{\mathrm e}^{2 t} \cos \left (3 t \right )}{145}+\frac {48 \,{\mathrm e}^{2 t} \sin \left (3 t \right )}{145}+\frac {4 \cos \left (2 t \right )}{145}-\frac {68 \sin \left (2 t \right )}{145} \\ y \left (t \right ) &= -\frac {141 \,{\mathrm e}^{2 t} \sin \left (3 t \right )}{145}+\frac {48 \,{\mathrm e}^{2 t} \cos \left (3 t \right )}{145}-\frac {54 \sin \left (2 t \right )}{145}-\frac {48 \cos \left (2 t \right )}{145} \\ \end{align*}

✓ Mathematica. Time used: 0.08 (sec). Leaf size: 86

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

\begin{align*} x(t)&\to \frac {1}{145} \left (-68 \sin (2 t)+48 e^{2 t} \sin (3 t)+4 \cos (2 t)+141 e^{2 t} \cos (3 t)\right )\\ y(t)&\to -\frac {3}{145} \left (18 \sin (2 t)+47 e^{2 t} \sin (3 t)+16 \cos (2 t)-16 e^{2 t} \cos (3 t)\right ) \end{align*}

✓ Sympy. Time used: 0.249 (sec). Leaf size: 68

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

\[ \left [ x{\left (t \right )} = \frac {33 e^{5 t}}{58} + \frac {4 \sin {\left (2 t \right )}}{145} - \frac {68 \cos {\left (2 t \right )}}{145} + \frac {9 e^{- t}}{10}, \ y{\left (t \right )} = - \frac {33 e^{5 t}}{58} + \frac {54 \sin {\left (2 t \right )}}{145} - \frac {48 \cos {\left (2 t \right )}}{145} + \frac {9 e^{- t}}{10}\right ] \]