58.16.14 problem 14

Internal problem ID [14898]
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 : 14
Date solved : Thursday, October 02, 2025 at 09:56:22 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.136 (sec). Leaf size: 41

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

\begin{align*} x \left (t \right ) &= \sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 -3-t \\ y \left (t \right ) &= -\cos \left (t \right ) c_2 +\sin \left (t \right ) c_1 -1-\sin \left (t \right ) c_2 -\cos \left (t \right ) c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 44

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

\begin{align*} x(t)&\to -t+c_1 \cos (t)-(c_1+c_2) \sin (t)-3\\ y(t)&\to c_2 \cos (t)+(2 c_1+c_2) \sin (t)-1 \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 73

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

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