58.16.10 problem 10

Internal problem ID [14894]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.1. Exercises page 277
Problem number : 10
Date solved : Thursday, October 02, 2025 at 09:56:18 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.129 (sec). Leaf size: 39

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} c_2 +{\mathrm e}^{3 t} c_1 -\frac {t}{6}-\frac {13}{18} \\ y \left (t \right ) &= \frac {{\mathrm e}^{t} c_2}{3}-{\mathrm e}^{3 t} c_1 -\frac {5}{18}+\frac {t}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.094 (sec). Leaf size: 74

ode={D[x[t],t]-D[y[t],t]-2*x[t]+4*y[t]==t,D[x[t],t]+D[y[t],t]-x[t]-y[t]==1}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{36} \left (-6 t+9 (c_1-3 c_2) e^{3 t}+27 (c_1+c_2) e^t-26\right )\\ y(t)&\to \frac {1}{36} \left (6 t-9 (c_1-3 c_2) e^{3 t}+9 (c_1+c_2) e^t-10\right ) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 42

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

\[ \left [ x{\left (t \right )} = 3 C_{1} e^{t} - C_{2} e^{3 t} - \frac {t}{6} - \frac {13}{18}, \ y{\left (t \right )} = C_{1} e^{t} + C_{2} e^{3 t} + \frac {t}{6} - \frac {5}{18}\right ] \]