90.19.10 problem 13

Internal problem ID [25292]
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 : 13
Date solved : Thursday, October 02, 2025 at 11:59:38 PM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )-y_{1} \left (t \right )&=-2 y_{2} \left (t \right )\\ y_{2}^{\prime }\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 )&=2 \\ y_{2} \left (0\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.060 (sec). Leaf size: 40

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 38

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

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

✓ Sympy. Time used: 0.072 (sec). Leaf size: 44

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

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