78.4.11 problem 12.c

Internal problem ID [21039]
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.c
Date solved : Thursday, October 02, 2025 at 07:01:41 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=3 \\ y \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.343 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= -\frac {11 \,{\mathrm e}^{t} \sin \left (t \right )}{5}+\frac {27 \,{\mathrm e}^{t} \cos \left (t \right )}{5}-\frac {\sin \left (t \right )}{5}-\frac {12 \cos \left (t \right )}{5} \\ y \left (t \right ) &= \frac {11 \,{\mathrm e}^{t} \cos \left (t \right )}{5}+\frac {27 \,{\mathrm e}^{t} \sin \left (t \right )}{5}-\frac {\cos \left (t \right )}{5}-\frac {13 \sin \left (t \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 60

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

\begin{align*} x(t)&\to \frac {1}{5} \left (3 \left (9 e^t-4\right ) \cos (t)-\left (11 e^t+1\right ) \sin (t)\right )\\ y(t)&\to \frac {1}{5} \left (\left (27 e^t-13\right ) \sin (t)+\left (11 e^t-1\right ) \cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.317 (sec). Leaf size: 75

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

\[ \left [ x{\left (t \right )} = - e^{t} \sin {\left (t \right )} + 3 e^{t} \cos {\left (t \right )} + \sin ^{3}{\left (t \right )} + \sin {\left (t \right )} \cos ^{2}{\left (t \right )}, \ y{\left (t \right )} = 3 e^{t} \sin {\left (t \right )} + e^{t} \cos {\left (t \right )} + \sin ^{3}{\left (t \right )} + \sin ^{2}{\left (t \right )} \cos {\left (t \right )} + \sin {\left (t \right )} \cos ^{2}{\left (t \right )} + \cos ^{3}{\left (t \right )}\right ] \]