70.19.24 problem 24

Internal problem ID [19034]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 24
Date solved : Thursday, October 02, 2025 at 03:37:08 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.262 (sec). Leaf size: 56

ode:=[diff(y__1(t),t) = y__2(t)-y__3(t), diff(y__2(t),t) = y__1(t)+y__3(t)-exp(-t), diff(y__3(t),t) = y__1(t)+y__2(t)+exp(t)]; 
ic:=[y__1(0) = 1, y__2(0) = 2, y__3(0) = 3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-t} t +\frac {3 \,{\mathrm e}^{-t}}{2}-\frac {{\mathrm e}^{t}}{2} \\ y_{2} \left (t \right ) &= -{\mathrm e}^{-t} t -{\mathrm e}^{-t}+3 \,{\mathrm e}^{t} \\ y_{3} \left (t \right ) &= -\frac {{\mathrm e}^{-t}}{2}+\frac {7 \,{\mathrm e}^{t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.098 (sec). Leaf size: 108

ode={D[y1[t],t]==y2[t]-y3[t],D[y2[t],t]==y1[t]+y3[t]-Exp[-t],D[y3[t],t]==y1[t]+y2[t]+Exp[t]}; 
ic={y1[0]==1,y2[0]==2,y3[0]==3}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to \log \left (e^t\right )-\frac {1}{2} e^{-t} \left (2 e^t t-2 t+e^{2 t}-3\right )\\ \text {y2}(t)&\to -e^t (t-3)-e^{-t}-e^{-t} t+t+\left (e^t-1\right ) \log \left (e^t\right )\\ \text {y3}(t)&\to e^t \left (\frac {7}{2}-t\right )-\frac {e^{-t}}{2}+t+\left (e^t-1\right ) \log \left (e^t\right ) \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-y__2(t) + y__3(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) - y__3(t) + Derivative(y__2(t), t) + exp(-t),0),Eq(-y__1(t) - y__2(t) - exp(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

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