76.19.19 problem 19

Internal problem ID [17664]
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 : 19
Date solved : Thursday, March 13, 2025 at 10:46:10 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=2 y_{1} \left (t \right )-64 y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=y_{1} \left (t \right )-14 y_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 25

ode:=[diff(y__1(t),t) = 2*y__1(t)-64*y__2(t), diff(y__2(t),t) = y__1(t)-14*y__2(t)]; 
ic:=y__1(0) = 0y__2(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= -64 \,{\mathrm e}^{-6 t} t \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-6 t} \left (-512 t +64\right )}{64} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 27

ode={D[y1[t],t]==2*y1[t]-64*y2[t],D[y2[t],t]==1*y1[t]-14*y2[t]}; 
ic={y1[0]==0,y2[0]==1}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to -64 e^{-6 t} t \\ \text {y2}(t)\to e^{-6 t} (1-8 t) \\ \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-2*y__1(t) + 64*y__2(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + 14*y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = 8 C_{1} t e^{- 6 t} + \left (C_{1} + 8 C_{2}\right ) e^{- 6 t}, \ y^{2}{\left (t \right )} = C_{1} t e^{- 6 t} + C_{2} e^{- 6 t}\right ] \]