90.19.4 problem 4

Internal problem ID [25286]
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 : 4
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 )+2 \frac {d}{d t}y_{2} \left (t \right )+y_{2} \left (t \right )&=-2 y_{1} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 37

ode:=[diff(y__1(t),t)-2*y__1(t) = 2*y__2(t), diff(diff(y__2(t),t),t)+2*diff(y__2(t),t)+y__2(t) = -2*y__1(t)]; 
ic:=[y__1(0) = 3, y__2(0) = 0, D(y__2)(0) = 3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-2 t}+2 \,{\mathrm e}^{t}+6 \,{\mathrm e}^{t} t \\ y_{2} \left (t \right ) &= -2 \,{\mathrm e}^{-2 t}+2 \,{\mathrm e}^{t}-3 \,{\mathrm e}^{t} t \\ \end{align*}

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 40

ode={D[y1[t],t]-2*y1[t]==2*y2[t], D[y2[t],{t,2}]+2*D[y2[t],t]+y2[t]==-2*y1[t]}; 
ic={y1[0]==3,y2[0]==0,Derivative[1][y2][0] ==3}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to e^t (6 t+2)+e^{-2 t}\\ \text {y2}(t)&\to e^t (2-3 t)-2 e^{-2 t} \end{align*}

✓ Sympy. Time used: 0.128 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-2*y1(t) - 2*y2(t) + Derivative(y1(t), t),0),Eq(2*y1(t) + y2(t) + 2*Derivative(y2(t), t) + Derivative(y2(t), (t, 2)),0)] 
ics = {y1(0): 3, y2(0): 0, Subs(Derivative(y2(t), t), t, 0): 3} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = 6 t e^{t} + 2 e^{t} + e^{- 2 t}, \ y_{2}{\left (t \right )} = - 3 t e^{t} + 2 e^{t} - 2 e^{- 2 t}\right ] \]