90.34.17 problem 17

Internal problem ID [25501]
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 : 17
Date solved : Friday, October 03, 2025 at 12:02:11 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} y_{1} \left (0\right )&=0 \\ y_{2} \left (0\right )&=0 \\ y_{3} \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 39

ode:=[diff(y__1(t),t) = y__2(t)+y__3(t)+exp(2*t), diff(y__2(t),t) = y__1(t)+y__2(t)-y__3(t)+exp(2*t), diff(y__3(t),t) = -2*y__1(t)+y__2(t)+3*y__3(t)-exp(2*t)]; 
ic:=[y__1(0) = 0, y__2(0) = 0, y__3(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= t \,{\mathrm e}^{t} \\ y_{2} \left (t \right ) &= \left (-1+2 t \right ) {\mathrm e}^{2 t}+{\mathrm e}^{t} \\ y_{3} \left (t \right ) &= t \,{\mathrm e}^{t}-2 \,{\mathrm e}^{2 t} t \\ \end{align*}

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 44

ode={D[y1[t],t]==y2[t]+y3[t]+Exp[2*t], D[y2[t],t]==y1[t]+y2[t]-y3[t]+Exp[2*t],D[y3[t],t]==-2*y1[t]+y2[t]+3*y3[t]-Exp[2*t]}; 
ic={y1[0]==0,y2[0]==0,y3[0]==0}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to e^t t\\ \text {y2}(t)&\to e^t \left (e^t (2 t-1)+1\right )\\ \text {y3}(t)&\to -e^t \left (2 e^t-1\right ) t \end{align*}

✓ Sympy. Time used: 0.169 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
y3 = Function("y3") 
ode=[Eq(-y2(t) - y3(t) - exp(2*t) + Derivative(y1(t), t),0),Eq(-y1(t) - y2(t) + y3(t) - exp(2*t) + Derivative(y2(t), t),0),Eq(2*y1(t) - y2(t) - 3*y3(t) + exp(2*t) + Derivative(y3(t), t),0)] 
ics = {y1(0): 0, y2(0): 0, y3(0): 0} 
dsolve(ode,func=[y1(t),y2(t),y3(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = t e^{t}, \ y_{2}{\left (t \right )} = 2 t e^{2 t} - e^{2 t} + e^{t}, \ y_{3}{\left (t \right )} = - 2 t e^{2 t} + t e^{t}\right ] \]