75.32.4 problem 828

Internal problem ID [17210]
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 : 828
Date solved : Monday, March 31, 2025 at 03:44:44 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.169 (sec). Leaf size: 42

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

\begin{align*} x \left (t \right ) &= -{\mathrm e}^{t}+\frac {{\mathrm e}^{4 t} c_1}{2}+{\mathrm e}^{2 t} c_2 \\ y \left (t \right ) &= {\mathrm e}^{t}+\frac {{\mathrm e}^{4 t} c_1}{2}-{\mathrm e}^{2 t} c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 70

ode={D[x[t],t]==3*x[t]+y[t]+Exp[t],D[y[t],t]==x[t]+3*y[t]-Exp[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-c_2) e^t+(c_1+c_2) e^{3 t}-2\right ) \\ y(t)\to \frac {1}{2} e^t \left ((c_2-c_1) e^t+(c_1+c_2) e^{3 t}+2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.149 (sec). Leaf size: 37

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

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