90.34.9 problem 9

Internal problem ID [25493]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 9. Linear Systems of Differential Equations. Exercises at page 677
Problem number : 9
Date solved : Friday, October 03, 2025 at 12:02:06 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=4 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=-y_{1} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=y_{1} \left (t \right )+4 y_{2} \left (t \right )-y_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right )&=2 \\ y_{2} \left (0\right )&=1 \\ y_{3} \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.076 (sec). Leaf size: 47

ode:=[diff(y__1(t),t) = 4*y__2(t), diff(y__2(t),t) = -y__1(t), diff(y__3(t),t) = y__1(t)+4*y__2(t)-y__3(t)]; 
ic:=[y__1(0) = 2, y__2(0) = 1, y__3(0) = 2]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 44

ode={D[y1[t],t]==4*y2[t], D[y2[t],t]==-y1[t], D[y3[t],t]==y1[t]+4*y2[t]-y3[t]}; 
ic={y1[0]==2,y2[0]==1,y3[0]==2}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to 2 (\sin (2 t)+\cos (2 t))\\ \text {y2}(t)&\to \cos (2 t)-\sin (2 t)\\ \text {y3}(t)&\to 2 (\sin (2 t)+\cos (2 t)) \end{align*}

✓ Sympy. Time used: 0.079 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
y3 = Function("y3") 
ode=[Eq(-4*y2(t) + Derivative(y1(t), t),0),Eq(y1(t) + Derivative(y2(t), t),0),Eq(-y1(t) - 4*y2(t) + y3(t) + Derivative(y3(t), t),0)] 
ics = {y1(0): 2, y2(0): 1, y3(0): 2} 
dsolve(ode,func=[y1(t),y2(t),y3(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = 2 \sin {\left (2 t \right )} + 2 \cos {\left (2 t \right )}, \ y_{2}{\left (t \right )} = - \sin {\left (2 t \right )} + \cos {\left (2 t \right )}, \ y_{3}{\left (t \right )} = 2 \sin {\left (2 t \right )} + 2 \cos {\left (2 t \right )}\right ] \]