82.7.15 problem 34-21

Internal problem ID [21871]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 34. Simulataneous linear differential equations. Page 1118
Problem number : 34-21
Date solved : Thursday, October 02, 2025 at 08:03:07 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+y \left (t \right )+3 \,{\mathrm e}^{2 t}\\ \frac {d}{d t}y \left (t \right )&=-4 x \left (t \right )+2 y \left (t \right )+{\mathrm e}^{2 t} t \end{align*}

✓ Maple. Time used: 0.088 (sec). Leaf size: 51

ode:=[diff(x(t),t) = 2*x(t)+y(t)+3*exp(2*t), diff(y(t),t) = -4*x(t)+2*y(t)+t*exp(2*t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (4 \cos \left (2 t \right ) c_1 +4 \sin \left (2 t \right ) c_2 +t \right )}{4} \\ y \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (-11+8 \cos \left (2 t \right ) c_2 -8 \sin \left (2 t \right ) c_1 \right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 62

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

\begin{align*} x(t)&\to \frac {1}{4} e^{2 t} (t+4 c_1 \cos (2 t)+2 c_2 \sin (2 t))\\ y(t)&\to \frac {1}{4} e^{2 t} (4 c_2 \cos (2 t)-8 c_1 \sin (2 t)-11) \end{align*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(5*t - 2*x(t) + y(t) + Derivative(x(t), t),0),Eq(-3*x(t) - 6*y(t) + Derivative(y(t), t) + 4,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} e^{3 t} - \frac {C_{2} e^{5 t}}{3} + 2 t + 1, \ y{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{5 t} - t\right ] \]