70.19.21 problem 21

Internal problem ID [19031]
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 : 21
Date solved : Thursday, October 02, 2025 at 03:37:06 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.231 (sec). Leaf size: 55

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

\begin{align*} y_{1} \left (t \right ) &= -\frac {589 \,{\mathrm e}^{4 t}}{225}-\frac {82 \,{\mathrm e}^{4 t} t}{15}-\frac {4 \,{\mathrm e}^{-t}}{25}-\frac {2 \,{\mathrm e}^{t}}{9} \\ y_{2} \left (t \right ) &= \frac {641 \,{\mathrm e}^{4 t}}{225}-\frac {82 \,{\mathrm e}^{4 t} t}{15}+\frac {{\mathrm e}^{-t}}{25}-\frac {8 \,{\mathrm e}^{t}}{9} \\ \end{align*}

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 354

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

\begin{align*} \text {y1}(t)&\to e^{4 t} \left (t \int _1^0e^{-5 K[2]} \left (2 e^{2 K[2]} (K[2]+1)-K[2]\right )dK[2]-t \int _1^te^{-5 K[2]} \left (2 e^{2 K[2]} (K[2]+1)-K[2]\right )dK[2]-(t+1) \int _1^0e^{-5 K[1]} \left (\left (-1+2 e^{2 K[1]}\right ) K[1]+1\right )dK[1]+(t+1) \int _1^te^{-5 K[1]} \left (\left (-1+2 e^{2 K[1]}\right ) K[1]+1\right )dK[1]+t+3\right )\\ \text {y2}(t)&\to e^{4 t} \left (t \left (-\int _1^0e^{-5 K[1]} \left (\left (-1+2 e^{2 K[1]}\right ) K[1]+1\right )dK[1]\right )+t \int _1^te^{-5 K[1]} \left (\left (-1+2 e^{2 K[1]}\right ) K[1]+1\right )dK[1]+t \int _1^0e^{-5 K[2]} \left (2 e^{2 K[2]} (K[2]+1)-K[2]\right )dK[2]-t \int _1^te^{-5 K[2]} \left (2 e^{2 K[2]} (K[2]+1)-K[2]\right )dK[2]+\int _1^te^{-5 K[2]} \left (2 e^{2 K[2]} (K[2]+1)-K[2]\right )dK[2]-\int _1^0e^{-5 K[2]} \left (2 e^{2 K[2]} (K[2]+1)-K[2]\right )dK[2]+t+2\right ) \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-5*y__1(t) + y__2(t) + Derivative(y__1(t), t) - exp(-t),0),Eq(-y__1(t) - 3*y__2(t) - 2*exp(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = C_{1} t e^{4 t} + \left (C_{1} + C_{2}\right ) e^{4 t} - \frac {2 e^{t}}{9} - \frac {4 e^{- t}}{25}, \ y^{2}{\left (t \right )} = C_{1} t e^{4 t} + C_{2} e^{4 t} - \frac {8 e^{t}}{9} + \frac {e^{- t}}{25}\right ] \]