75.32.3 problem 827

Internal problem ID [17209]
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 : 827
Date solved : Monday, March 31, 2025 at 03:44:43 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-4 y \left (t \right )+1\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+5 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.130 (sec). Leaf size: 33

ode:=[diff(x(t),t) = 2*x(t)-4*y(t)+1, diff(y(t),t) = -x(t)+5*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} c_2 +{\mathrm e}^{6 t} c_1 -\frac {5}{6} \\ y \left (t \right ) &= \frac {{\mathrm e}^{t} c_2}{4}-{\mathrm e}^{6 t} c_1 -\frac {1}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 67

ode={D[x[t],t]==2*x[t]-4*y[t],D[y[t],t]==-x[t]+5*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 (e^{5 t}+4\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}\right )+4 c_2 e^{5 t}+c_1+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + 4*y(t) + Derivative(x(t), t) - 1,0),Eq(x(t) - 5*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 4 C_{1} e^{t} - C_{2} e^{6 t} - \frac {5}{6}, \ y{\left (t \right )} = C_{1} e^{t} + C_{2} e^{6 t} - \frac {1}{6}\right ] \]