76.19.16 problem 16

Internal problem ID [17661]
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 : 16
Date solved : Thursday, March 13, 2025 at 10:46:07 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=4 y_{1} \left (t \right )-4 y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=5 y_{1} \left (t \right )-4 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: 0.097 (sec). Leaf size: 24

ode:=[diff(y__1(t),t) = 4*y__1(t)-4*y__2(t), diff(y__2(t),t) = 5*y__1(t)-4*y__2(t)]; 
ic:=y__1(0) = 1y__2(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= \cos \left (2 t \right )+2 \sin \left (2 t \right ) \\ y_{2} \left (t \right ) &= \frac {5 \sin \left (2 t \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 25

ode={D[y1[t],t]==4*y1[t]-4*y2[t],D[y2[t],t]==5*y1[t]-4*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 2 \sin (2 t)+\cos (2 t) \\ \text {y2}(t)\to 5 \sin (t) \cos (t) \\ \end{align*}

✓ Sympy. Time used: 0.087 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-4*y__1(t) + 4*y__2(t) + Derivative(y__1(t), t),0),Eq(-5*y__1(t) + 4*y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = - \left (\frac {2 C_{1}}{5} - \frac {4 C_{2}}{5}\right ) \cos {\left (2 t \right )} - \left (\frac {4 C_{1}}{5} + \frac {2 C_{2}}{5}\right ) \sin {\left (2 t \right )}, \ y^{2}{\left (t \right )} = - C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )}\right ] \]