69.1.133 problem 192

Internal problem ID [14207]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 192
Date solved : Wednesday, March 05, 2025 at 10:40:04 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-4 x \left (t \right )-10 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 58

ode:=[diff(x(t),t) = -4*x(t)-10*y(t), diff(y(t),t) = x(t)-2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-3 t} \left (c_{1} \sin \left (3 t \right )-3 \sin \left (3 t \right ) c_{2} +3 \cos \left (3 t \right ) c_{1} +c_{2} \cos \left (3 t \right )\right )}{10} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 67

ode={D[x[t],t]==-4*x[t]-10*y[t],D[y[t],t]==x[t]-2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{-3 t} (3 c_1 \cos (3 t)-(c_1+10 c_2) \sin (3 t)) \\ y(t)\to \frac {1}{3} e^{-3 t} (3 c_2 \cos (3 t)+(c_1+c_2) \sin (3 t)) \\ \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(4*x(t) + 10*y(t) + Derivative(x(t), t),0),Eq(-x(t) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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