54.9.8 problem 1863

Internal problem ID [13086]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1863
Date solved : Wednesday, October 01, 2025 at 03:34:22 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+3 x \left (t \right )+4 y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )+2 x \left (t \right )+5 y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.108 (sec). Leaf size: 34

ode:={diff(x(t),t)+3*x(t)+4*y(t) = 0, diff(y(t),t)+2*x(t)+5*y(t) = 0}; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 72

ode={D[x[t],t]+3*x[t]+4*y[t]==0,D[y[t],t]+2*x[t]+5*y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{3} e^{-7 t} \left (c_1 \left (2 e^{6 t}+1\right )-2 c_2 \left (e^{6 t}-1\right )\right )\\ y(t)&\to \frac {1}{3} e^{-7 t} \left (c_2 \left (e^{6 t}+2\right )-c_1 \left (e^{6 t}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.049 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) + 4*y(t) + Derivative(x(t), t),0),Eq(2*x(t) + 5*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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