58.16.8 problem 8

Internal problem ID [14892]
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 : 8
Date solved : Thursday, October 02, 2025 at 09:56:17 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.132 (sec). Leaf size: 59

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\sqrt {3}\, t} c_2 +{\mathrm e}^{-\sqrt {3}\, t} c_1 +3 t -3 \\ y \left (t \right ) &= \frac {\sqrt {3}\, {\mathrm e}^{\sqrt {3}\, t} c_2}{3}-\frac {\sqrt {3}\, {\mathrm e}^{-\sqrt {3}\, t} c_1}{3}+\frac {4}{3}-2 t \\ \end{align*}

✓ Mathematica. Time used: 2.77 (sec). Leaf size: 137

ode={D[x[t],t]+D[y[t],t]-x[t]-3*y[t]==3*t,D[x[t],t]+2*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 \frac {1}{2} e^{-\sqrt {3} t} \left (6 e^{\sqrt {3} t} (t-1)+\left (c_1+\sqrt {3} c_2\right ) e^{2 \sqrt {3} t}+c_1-\sqrt {3} c_2\right )\\ y(t)&\to \frac {1}{6} e^{-\sqrt {3} t} \left (e^{\sqrt {3} t} (8-12 t)+\left (\sqrt {3} c_1+3 c_2\right ) e^{2 \sqrt {3} t}-\sqrt {3} c_1+3 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.279 (sec). Leaf size: 66

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

\[ \left [ x{\left (t \right )} = \sqrt {3} C_{1} e^{\sqrt {3} t} - \sqrt {3} C_{2} e^{- \sqrt {3} t} + 3 t - 3, \ y{\left (t \right )} = C_{1} e^{\sqrt {3} t} + C_{2} e^{- \sqrt {3} t} - 2 t + \frac {4}{3}\right ] \]