90.34.12 problem 12

Internal problem ID [25496]
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 : 12
Date solved : Friday, October 03, 2025 at 12:02:08 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )-5 y_{2} \left (t \right )+2 \cos \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-2 y_{2} \left (t \right )+\cos \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: 0.097 (sec). Leaf size: 37

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

\begin{align*} y_{1} \left (t \right ) &= 8 \sin \left (t \right )+\cos \left (t \right )-\frac {\sin \left (t \right ) t}{2}+\cos \left (t \right ) t \\ y_{2} \left (t \right ) &= -\cos \left (t \right )+\frac {7 \sin \left (t \right )}{2}+\frac {\cos \left (t \right ) t}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 39

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

\begin{align*} \text {y1}(t)&\to (t+1) \cos (t)-\frac {1}{2} (t-16) \sin (t)\\ \text {y2}(t)&\to \frac {1}{2} (7 \sin (t)+(t-2) \cos (t)) \end{align*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 70

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-2*y1(t) + 5*y2(t) - 2*cos(t) + Derivative(y1(t), t),0),Eq(-y1(t) + 2*y2(t) - cos(t) + Derivative(y2(t), t),0)] 
ics = {y1(0): 1, y2(0): -1} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = - \frac {t \sin {\left (t \right )}}{2} + t \cos {\left (t \right )} + \sin ^{3}{\left (t \right )} + \sin {\left (t \right )} \cos ^{2}{\left (t \right )} + 7 \sin {\left (t \right )} + \cos {\left (t \right )}, \ y_{2}{\left (t \right )} = \frac {t \cos {\left (t \right )}}{2} + \frac {\sin ^{3}{\left (t \right )}}{2} + \frac {\sin {\left (t \right )} \cos ^{2}{\left (t \right )}}{2} + 3 \sin {\left (t \right )} - \cos {\left (t \right )}\right ] \]