78.5.14 problem 7.b

Internal problem ID [21053]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 7.b
Date solved : Thursday, October 02, 2025 at 07:01:48 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=12 x \left (t \right )-15 y\\ y^{\prime }&=4 x \left (t \right )-4 y \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.119 (sec). Leaf size: 31

ode:=[diff(x(t),t) = 12*x(t)-15*y(t), diff(y(t),t) = 4*x(t)-4*y(t)]; 
ic:=[x(0) = 1, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {5 \,{\mathrm e}^{6 t}}{2}-\frac {3 \,{\mathrm e}^{2 t}}{2} \\ y \left (t \right ) &= {\mathrm e}^{6 t}-{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 39

ode={D[x[t],t]==12*x[t]-15*y[t],D[y[t],t]==4*x[t]-4*y[t]}; 
ic={x[0]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} e^{2 t} \left (5 e^{4 t}-3\right )\\ y(t)&\to e^{2 t} \left (e^{4 t}-1\right ) \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-12*x(t) + 15*y(t) + Derivative(x(t), t),0),Eq(-4*x(t) + 4*y(t) + Derivative(y(t), t),0)] 
ics = {x(0): 1, y(0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {5 e^{6 t}}{2} - \frac {3 e^{2 t}}{2}, \ y{\left (t \right )} = e^{6 t} - e^{2 t}\right ] \]