90.19.6 problem 6

Internal problem ID [25288]
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 : 6
Date solved : Sunday, October 12, 2025 at 05:55:37 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )-2 y_{1} \left (t \right )&=-2 y_{2} \left (t \right )\\ \frac {d^{2}}{d t^{2}}y_{2} \left (t \right )+\frac {d}{d t}y_{2} \left (t \right )+6 y_{2} \left (t \right )&=4 y_{1} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.063 (sec). Leaf size: 41

ode:=[diff(y__1(t),t)-2*y__1(t) = -2*y__2(t), diff(diff(y__2(t),t),t)+diff(y__2(t),t)+6*y__2(t) = 4*y__1(t)]; 
ic:=[y__1(0) = 1, y__2(0) = 5, D(y__2)(0) = 4]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= -4 \,{\mathrm e}^{t}-2 \sin \left (2 t \right )+5 \cos \left (2 t \right ) \\ y_{2} \left (t \right ) &= -2 \,{\mathrm e}^{t}+3 \sin \left (2 t \right )+7 \cos \left (2 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 44

ode={D[y1[t],t]-2*y1[t]==-2*y2[t], D[y2[t],{t,2}]+D[y2[t],t]+6*y2[t]==4*y1[t]}; 
ic={y1[0]==1,y2[0]==5,Derivative[1][y2][0] ==4}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to 5 \cos (2 t)-4 \left (e^t+\sin (t) \cos (t)\right )\\ \text {y2}(t)&\to -2 e^t+3 \sin (2 t)+7 \cos (2 t) \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(4*y1(t) - 10*y2(t) + Derivative(y1(t), t),0),Eq(-9*y1(t) + 23*y2(t) - 6*Derivative(y2(t), t) + Derivative(y2(t), (t, 2)),0)] 
ics = {y1(0): 0, y2(0): 2, Subs(Derivative(y2(t), t), t, 0): 2} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = - 20 e^{2 t} + 40 e^{t} - 20 e^{- t}, \ y_{2}{\left (t \right )} = - 12 e^{2 t} + 20 e^{t} - 6 e^{- t}\right ] \]