90.34.2 problem 2

Internal problem ID [25486]
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 : 2
Date solved : Friday, October 03, 2025 at 12:02:02 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.058 (sec). Leaf size: 49

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

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 58

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

\begin{align*} \text {y1}(t)&\to \cos \left (\sqrt {2} t\right )-\frac {\sin \left (\sqrt {2} t\right )}{\sqrt {2}}\\ \text {y2}(t)&\to -\sqrt {2} \sin \left (\sqrt {2} t\right )-\cos \left (\sqrt {2} t\right ) \end{align*}

✓ Sympy. Time used: 0.099 (sec). Leaf size: 51

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

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