90.34.15 problem 15

Internal problem ID [25499]
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 : 15
Date solved : Friday, October 03, 2025 at 12:02:10 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=5 y_{1} \left (t \right )+2 y_{2} \left (t \right )+t\\ y_{2}^{\prime }\left (t \right )&=-8 y_{1} \left (t \right )-3 y_{2} \left (t \right )-2 t \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 33

ode:=[diff(y__1(t),t) = 5*y__1(t)+2*y__2(t)+t, diff(y__2(t),t) = -8*y__1(t)-3*y__2(t)-2*t]; 
ic:=[y__1(0) = 0, y__2(0) = 1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{t}+2 \,{\mathrm e}^{t} t -t -1 \\ y_{2} \left (t \right ) &= -{\mathrm e}^{t}-4 \,{\mathrm e}^{t} t +2+2 t \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 38

ode={D[y1[t],t]==5*y1[t]+2*y2[t]+t, D[y2[t],t]==-8*y1[t]-3*y2[t]-2*t}; 
ic={y1[0]==0,y2[0]==1}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to -t+e^t (2 t+1)-1\\ \text {y2}(t)&\to 2 (t+1)-e^t (4 t+1) \end{align*}

✓ Sympy. Time used: 0.114 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-t - 5*y1(t) - 2*y2(t) + Derivative(y1(t), t),0),Eq(2*t + 8*y1(t) + 3*y2(t) + Derivative(y2(t), t),0)] 
ics = {y1(0): 0, y2(0): 1} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = 2 t e^{t} - t + e^{t} - 1, \ y_{2}{\left (t \right )} = - 4 t e^{t} + 2 t - e^{t} + 2\right ] \]