90.19.7 problem 7

Internal problem ID [25289]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 4. Linear Constant Coefficient Differential Equations. Exercises at page 309
Problem number : 7
Date solved : Sunday, October 12, 2025 at 05:55:38 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}y_{1} \left (t \right )+2 \frac {d}{d t}y_{1} \left (t \right )+6 y_{1} \left (t \right )&=5 y_{2} \left (t \right )\\ \frac {d^{2}}{d t^{2}}y_{2} \left (t \right )-2 \frac {d}{d t}y_{2} \left (t \right )+6 y_{2} \left (t \right )&=9 y_{1} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right )&=0 \\ D\left (y_{1} \right )\left (0\right )&=0 \\ y_{2} \left (0\right )&=6 \\ D\left (y_{2} \right )\left (0\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 43

ode:=[diff(diff(y__1(t),t),t)+2*diff(y__1(t),t)+6*y__1(t) = 5*y__2(t), diff(diff(y__2(t),t),t)-2*diff(y__2(t),t)+6*y__2(t) = 9*y__1(t)]; 
ic:=[y__1(0) = 0, D(y__1)(0) = 0, y__2(0) = 6, D(y__2)(0) = 6]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 42

ode={D[y1[t],{t,2}]+2*D[y1[t],t]+6*y1[t]==5*y2[t], D[y2[t],{t,2}]-2*D[y2[t],t]+6*y2[t]==9*y1[t]}; 
ic={y1[0]==0,Derivative[1][y1][0] ==0,y2[0]==6,Derivative[1][y2][0] ==6}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to 3 e^{-t}+\sin (3 t)-3 \cos (3 t)\\ \text {y2}(t)&\to 3 \left (e^{-t}+\sin (3 t)+\cos (3 t)\right ) \end{align*}

✓ Sympy. Time used: 0.147 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(6*y1(t) - 5*y2(t) + 2*Derivative(y1(t), t) + Derivative(y1(t), (t, 2)),0),Eq(-9*y1(t) + 6*y2(t) - 2*Derivative(y2(t), t) + Derivative(y2(t), (t, 2)),0)] 
ics = {y1(0): 0, Subs(Derivative(y1(t), t), t, 0): 0, y2(0): 6, Subs(Derivative(y2(t), t), t, 0): 6} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

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