85.88.14 problem 2 (d)

Internal problem ID [23041]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 499
Problem number : 2 (d)
Date solved : Thursday, October 02, 2025 at 09:17:44 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+3 x \left (t \right )-\frac {d}{d t}y \left (t \right )-y \left (t \right )&=0\\ 2 \frac {d}{d t}x \left (t \right )-9 x \left (t \right )+\frac {d}{d t}y \left (t \right )+4 y \left (t \right )&=15 \,{\mathrm e}^{-3 t} \end{align*}

✓ Maple. Time used: 0.089 (sec). Leaf size: 42

ode:=[diff(x(t),t)+3*x(t)-diff(y(t),t)-y(t) = 0, 2*diff(x(t),t)-9*x(t)+diff(y(t),t)+4*y(t) = 15*exp(-3*t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 50

ode={D[x[t],{t,1}]+3*x[t]-D[y[t],t]-y[t]==0, 2*D[x[t],t]-9*x[t]+ D[y[t],{t,1}]+4*y[t]==15*Exp[-3*t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -e^{-3 t}+c_1 \cos (t)+(2 c_1-c_2) \sin (t)\\ y(t)&\to 5 c_1 \sin (t)+c_2 (\cos (t)-2 \sin (t)) \end{align*}

✓ Sympy. Time used: 0.236 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) - y(t) + Derivative(x(t), t) - Derivative(y(t), t),0),Eq(-9*x(t) + 4*y(t) + 2*Derivative(x(t), t) + Derivative(y(t), t) - 15*exp(-3*t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {C_{1}}{5} - \frac {2 C_{2}}{5}\right ) \cos {\left (t \right )} - \left (\frac {2 C_{1}}{5} + \frac {C_{2}}{5}\right ) \sin {\left (t \right )} - e^{- 3 t} \sin ^{2}{\left (t \right )} - e^{- 3 t} \cos ^{2}{\left (t \right )}, \ y{\left (t \right )} = - C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )}\right ] \]