76.19.15 problem 15

Internal problem ID [17660]
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 : 15
Date solved : Thursday, March 13, 2025 at 10:46:06 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=5 y_{1} \left (t \right )-2 y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=6 y_{1} \left (t \right )-2 y_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.191 (sec). Leaf size: 29

ode:=[diff(y__1(t),t) = 5*y__1(t)-2*y__2(t), diff(y__2(t),t) = 6*y__1(t)-2*y__2(t)]; 
ic:=y__1(0) = 1y__2(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= 4 \,{\mathrm e}^{2 t}-3 \,{\mathrm e}^{t} \\ y_{2} \left (t \right ) &= 6 \,{\mathrm e}^{2 t}-6 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 29

ode={D[y1[t],t]==5*y1[t]-2*y2[t],D[y2[t],t]==6*y1[t]-2*y2[t]}; 
ic={y1[0]==1,y2[0]==0}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to e^t \left (4 e^t-3\right ) \\ \text {y2}(t)\to 6 e^t \left (e^t-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.080 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-5*y__1(t) + 2*y__2(t) + Derivative(y__1(t), t),0),Eq(-6*y__1(t) + 2*y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

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