90.33.10 problem 10

Internal problem ID [25484]
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 645
Problem number : 10
Date solved : Friday, October 03, 2025 at 12:02:01 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 26

ode={D[y1[t],t]==y2[t]+t, D[y2[t],t]==-y1[t]-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+2 \sin (t)+1\\ \text {y2}(t)&\to -t+2 \cos (t)-1 \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 61

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

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