87.21.15 problem 15

Internal problem ID [23729]
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 : 15
Date solved : Thursday, October 02, 2025 at 09:44:39 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 31

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

\begin{align*} x \left (t \right ) &= -3 c_1 \,{\mathrm e}^{3 t}-c_2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}+c_2 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 67

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

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

✓ Sympy. Time used: 0.054 (sec). Leaf size: 31

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

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