30.5.34 problem 45

Internal problem ID [7533]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.6, Substitutions and Transformations. Exercises. page 76
Problem number : 45
Date solved : Tuesday, September 30, 2025 at 04:44:50 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y \left (t \right )&=-4 x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 35

ode:=[diff(y(t),t) = -4*x(t)-y(t), diff(x(t),t) = 2*x(t)-y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}+c_2 \,{\mathrm e}^{-2 t} \\ y \left (t \right ) &= -c_1 \,{\mathrm e}^{3 t}+4 c_2 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 71

ode={D[y[t],t]==-4*x[t]-y[t],D[x[t],t]==2*x[t]-y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{5} e^{-2 t} \left (4 c_1 e^{5 t}-c_2 e^{5 t}+c_1+c_2\right )\\ y(t)&\to \frac {1}{5} e^{-2 t} \left (c_2 \left (e^{5 t}+4\right )-4 c_1 \left (e^{5 t}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.050 (sec). Leaf size: 36

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

\[ \left [ x{\left (t \right )} = 2 C_{1} t e^{t} + \left (C_{1} + 2 C_{2}\right ) e^{t}, \ y{\left (t \right )} = 4 C_{1} t e^{t} + 4 C_{2} e^{t}\right ] \]