87.21.13 problem 13

Internal problem ID [23727]
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 : 13
Date solved : Thursday, October 02, 2025 at 09:44:38 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=-y \left (t \right )\\ y^{\prime }\left (t \right )&=-3 x+4 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 63

ode:=[diff(x(t),t) = -y(t), diff(y(t),t) = -3*x(t)+4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\left (2+\sqrt {7}\right ) t}+c_2 \,{\mathrm e}^{-\left (-2+\sqrt {7}\right ) t} \\ y \left (t \right ) &= \left (-2+\sqrt {7}\right ) c_2 \,{\mathrm e}^{-\left (-2+\sqrt {7}\right ) t}+\left (-2-\sqrt {7}\right ) c_1 \,{\mathrm e}^{\left (2+\sqrt {7}\right ) t} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 148

ode={D[x[t],t]==-y[t],D[y[t],t]==-3*x[t]+4*y[t] }; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{14} e^{-\left (\left (\sqrt {7}-2\right ) t\right )} \left (c_1 \left (\left (7-2 \sqrt {7}\right ) e^{2 \sqrt {7} t}+7+2 \sqrt {7}\right )-\sqrt {7} c_2 \left (e^{2 \sqrt {7} t}-1\right )\right )\\ y(t)&\to \frac {1}{14} e^{-\left (\left (\sqrt {7}-2\right ) t\right )} \left (c_2 \left (\left (7+2 \sqrt {7}\right ) e^{2 \sqrt {7} t}+7-2 \sqrt {7}\right )-3 \sqrt {7} c_1 \left (e^{2 \sqrt {7} t}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.158 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(y(t) + Derivative(x(t), t),0),Eq(3*x(t) - 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} \left (2 - \sqrt {7}\right ) e^{t \left (2 + \sqrt {7}\right )}}{3} + \frac {C_{2} \left (2 + \sqrt {7}\right ) e^{t \left (2 - \sqrt {7}\right )}}{3}, \ y{\left (t \right )} = C_{1} e^{t \left (2 + \sqrt {7}\right )} + C_{2} e^{t \left (2 - \sqrt {7}\right )}\right ] \]