57.19.7 problem 3(a)

Internal problem ID [14502]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 202
Problem number : 3(a)
Date solved : Thursday, October 02, 2025 at 09:37:55 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=2 x+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x-14 \end{align*}

✓ Maple. Time used: 0.162 (sec). Leaf size: 76

ode:=[diff(x(t),t) = 2*x(t)+3*y(t), diff(y(t),t) = -x(t)-14]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -14+{\mathrm e}^{t} \left (\sin \left (\sqrt {2}\, t \right ) \sqrt {2}\, c_1 -\cos \left (\sqrt {2}\, t \right ) \sqrt {2}\, c_2 -\sin \left (\sqrt {2}\, t \right ) c_2 -\cos \left (\sqrt {2}\, t \right ) c_1 \right ) \\ y \left (t \right ) &= \frac {28}{3}+{\mathrm e}^{t} \left (\sin \left (\sqrt {2}\, t \right ) c_2 +\cos \left (\sqrt {2}\, t \right ) c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.107 (sec). Leaf size: 89

ode={D[x[t],t]==2*x[t]+3*y[t],D[y[t],t]==-x[t]-14}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to c_1 e^t \cos \left (\sqrt {2} t\right )+\frac {(c_1+3 c_2) e^t \sin \left (\sqrt {2} t\right )}{\sqrt {2}}-14\\ y(t)&\to c_2 e^t \cos \left (\sqrt {2} t\right )-\frac {(c_1+c_2) e^t \sin \left (\sqrt {2} t\right )}{\sqrt {2}}+\frac {28}{3} \end{align*}

✓ Sympy. Time used: 0.238 (sec). Leaf size: 122

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(x(t) + Derivative(y(t), t) + 14,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (C_{1} + \sqrt {2} C_{2}\right ) e^{t} \sin {\left (\sqrt {2} t \right )} + \left (\sqrt {2} C_{1} - C_{2}\right ) e^{t} \cos {\left (\sqrt {2} t \right )} - 14 \sin ^{2}{\left (\sqrt {2} t \right )} - 14 \cos ^{2}{\left (\sqrt {2} t \right )}, \ y{\left (t \right )} = - C_{1} e^{t} \sin {\left (\sqrt {2} t \right )} + C_{2} e^{t} \cos {\left (\sqrt {2} t \right )} + \frac {28 \sin ^{2}{\left (\sqrt {2} t \right )}}{3} + \frac {28 \cos ^{2}{\left (\sqrt {2} t \right )}}{3}\right ] \]