76.19.17 problem 17

Internal problem ID [17662]
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 : 17
Date solved : Thursday, March 13, 2025 at 10:46:08 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=6 y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-6 y_{1} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 5\\ y_{2} \left (0\right ) = 4 \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 33

ode:=[diff(y__1(t),t) = 6*y__2(t), diff(y__2(t),t) = -6*y__1(t)]; 
ic:=y__1(0) = 5y__2(0) = 4; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= 4 \sin \left (6 t \right )+5 \cos \left (6 t \right ) \\ y_{2} \left (t \right ) &= 4 \cos \left (6 t \right )-5 \sin \left (6 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 34

ode={D[y1[t],t]==0*y1[t]+6*y2[t],D[y2[t],t]==-6*y1[t]+0*y2[t]}; 
ic={y1[0]==5,y2[0]==4}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to 4 \sin (6 t)+5 \cos (6 t) \\ \text {y2}(t)\to 4 \cos (6 t)-5 \sin (6 t) \\ \end{align*}

✓ Sympy. Time used: 0.065 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-6*y__2(t) + Derivative(y__1(t), t),0),Eq(6*y__1(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = C_{1} \sin {\left (6 t \right )} + C_{2} \cos {\left (6 t \right )}, \ y^{2}{\left (t \right )} = C_{1} \cos {\left (6 t \right )} - C_{2} \sin {\left (6 t \right )}\right ] \]