76.19.20 problem 20

Internal problem ID [17665]
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 : 20
Date solved : Thursday, March 13, 2025 at 10:46:11 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 2.717 (sec). Leaf size: 67

ode:=[diff(y__1(t),t) = -4*y__1(t)-y__2(t)+2*exp(t), diff(y__2(t),t) = y__1(t)-2*y__2(t)+sin(2*t)]; 
ic:=y__1(0) = 1y__2(0) = 2; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= \frac {749 \,{\mathrm e}^{-3 t}}{1352}-\frac {69 t \,{\mathrm e}^{-3 t}}{26}+\frac {3 \,{\mathrm e}^{t}}{8}+\frac {12 \cos \left (2 t \right )}{169}-\frac {5 \sin \left (2 t \right )}{169} \\ y_{2} \left (t \right ) &= \frac {2839 \,{\mathrm e}^{-3 t}}{1352}+\frac {69 t \,{\mathrm e}^{-3 t}}{26}+\frac {{\mathrm e}^{t}}{8}+\frac {44 \sin \left (2 t \right )}{169}-\frac {38 \cos \left (2 t \right )}{169} \\ \end{align*}

✓ Mathematica. Time used: 0.403 (sec). Leaf size: 94

ode={D[y1[t],t]==-4*y1[t]-1*y2[t]+2*Exp[t],D[y2[t],t]==1*y1[t]-2*y2[t]+Sin[2*t]}; 
ic={y1[0]==1,y2[0]==2}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to \frac {e^{-3 t} \left (-3588 t+507 e^{4 t}-40 e^{3 t} \sin (2 t)+96 e^{3 t} \cos (2 t)+749\right )}{1352} \\ \text {y2}(t)\to \frac {e^{-3 t} \left (3588 t+169 e^{4 t}+352 e^{3 t} \sin (2 t)-304 e^{3 t} \cos (2 t)+2839\right )}{1352} \\ \end{align*}

✓ Sympy. Time used: 1.748 (sec). Leaf size: 82

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(4*y__1(t) + y__2(t) - 2*exp(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + 2*y__2(t) - sin(2*t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = - C_{2} t e^{- 3 t} - \left (C_{1} - C_{2}\right ) e^{- 3 t} + \frac {3 e^{t}}{8} - \frac {5 \sin {\left (2 t \right )}}{169} + \frac {12 \cos {\left (2 t \right )}}{169}, \ y^{2}{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} t e^{- 3 t} + \frac {e^{t}}{8} + \frac {44 \sin {\left (2 t \right )}}{169} - \frac {38 \cos {\left (2 t \right )}}{169}\right ] \]