82.7.13 problem 34-19

Internal problem ID [21869]
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 : 34-19
Date solved : Thursday, October 02, 2025 at 08:03:05 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )+{\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+3 y \left (t \right )+1 \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 42

ode:=[diff(x(t),t) = y(t)+exp(t), diff(y(t),t) = -2*x(t)+3*y(t)+1]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} c_1 +\frac {1}{2}+2 \ln \left ({\mathrm e}^{t}\right ) {\mathrm e}^{t}+c_2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= {\mathrm e}^{t} \left (2 \,{\mathrm e}^{t} c_1 +2 \ln \left ({\mathrm e}^{t}\right )+c_2 +1\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 71

ode={D[x[t],t]==y[t]+Exp[t],D[y[t],t]==-2*x[t]+3*y[t]+1}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.094 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) - exp(t) + Derivative(x(t), t),0),Eq(2*x(t) - 3*y(t) + Derivative(y(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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