76.19.14 problem 14

Internal problem ID [17659]
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 : 14
Date solved : Thursday, March 13, 2025 at 10:46:05 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-5 y_{1} \left (t \right )+y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-9 y_{1} \left (t \right )+5 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: 2.509 (sec). Leaf size: 33

ode:=[diff(y__1(t),t) = -5*y__1(t)+y__2(t), diff(y__2(t),t) = -9*y__1(t)+5*y__2(t)]; 
ic:=y__1(0) = 1y__2(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= \frac {9 \,{\mathrm e}^{-4 t}}{8}-\frac {{\mathrm e}^{4 t}}{8} \\ y_{2} \left (t \right ) &= \frac {9 \,{\mathrm e}^{-4 t}}{8}-\frac {9 \,{\mathrm e}^{4 t}}{8} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 40

ode={D[y1[t],t]==-5*y1[t]+y2[t],D[y2[t],t]==-9*y1[t]+5*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 -\frac {1}{8} e^{-4 t} \left (e^{8 t}-9\right ) \\ \text {y2}(t)\to -\frac {9}{8} e^{-4 t} \left (e^{8 t}-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.099 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(5*y__1(t) - y__2(t) + Derivative(y__1(t), t),0),Eq(9*y__1(t) - 5*y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = C_{1} e^{- 4 t} + \frac {C_{2} e^{4 t}}{9}, \ y^{2}{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{4 t}\right ] \]