70.19.18 problem 18

Internal problem ID [19028]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 18
Date solved : Thursday, October 02, 2025 at 03:37:05 PM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=-4 y_{1} \left (t \right )-y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-2 y_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right )&=1 \\ y_{2} \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.127 (sec). Leaf size: 23

ode:=[diff(y__1(t),t) = -4*y__1(t)-y__2(t), diff(y__2(t),t) = y__1(t)-2*y__2(t)]; 
ic:=[y__1(0) = 1, y__2(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (-t +1\right ) \\ y_{2} \left (t \right ) &= {\mathrm e}^{-3 t} t \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 25

ode={D[y1[t],t]==-4*y1[t]-y2[t],D[y2[t],t]==1*y1[t]-2*y2[t]}; 
ic={y1[0]==1,y2[0]==0}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to -e^{-3 t} (t-1)\\ \text {y2}(t)&\to e^{-3 t} t \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(4*y__1(t) + y__2(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + 2*y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

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