87.21.9 problem 9

Internal problem ID [23723]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 178
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:44:36 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=3 x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=12 x-7 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ y \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 33

ode:=[diff(x(t),t) = 3*x(t)-2*y(t), diff(y(t),t) = 12*x(t)-7*y(t)]; 
ic:=[x(0) = 0, y(0) = 3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{-3 t}-3 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 9 \,{\mathrm e}^{-3 t}-6 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 37

ode={D[x[t],t]==3*x[t]-2*y[t],D[y[t],t]==12*x[t]-7*y[t] }; 
ic={x[0]==0,y[0]==3}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -3 e^{-3 t} \left (e^{2 t}-1\right )\\ y(t)&\to e^{-3 t} \left (9-6 e^{2 t}\right ) \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-12*x(t) + 7*y(t) + Derivative(y(t), t),0)] 
ics = {x(0): 0, y(0): 3} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 3 e^{- t} + 3 e^{- 3 t}, \ y{\left (t \right )} = - 6 e^{- t} + 9 e^{- 3 t}\right ] \]