69.32.1 problem 825

Internal problem ID [18565]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 23.3 dAlemberts method. Exercises page 243
Problem number : 825
Date solved : Thursday, October 02, 2025 at 03:14:58 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )+4 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: 0.111 (sec). Leaf size: 31

ode:=[diff(x(t),t) = 5*x(t)+4*y(t), diff(y(t),t) = x(t)+2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{6 t}+c_2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{6 t}}{4}-c_2 \,{\mathrm e}^{t} \\ \end{align*}

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

ode={D[x[t],t]==5*x[t]+4*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}{5} e^t \left (c_1 \left (4 e^{5 t}+1\right )+4 c_2 \left (e^{5 t}-1\right )\right )\\ y(t)&\to \frac {1}{5} e^t \left (c_1 \left (e^{5 t}-1\right )+c_2 \left (e^{5 t}+4\right )\right ) \end{align*}

✓ Sympy. Time used: 0.043 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) - 4*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)
 

\[ \left [ x{\left (t \right )} = - C_{1} e^{t} + 4 C_{2} e^{6 t}, \ y{\left (t \right )} = C_{1} e^{t} + C_{2} e^{6 t}\right ] \]