64.16.14 problem 14

Internal problem ID [13546]
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 : Monday, March 31, 2025 at 08:00:59 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: 0.046 (sec). Leaf size: 158

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 (\cos (t)-\sin (t)) \int _1^t((K[1]+2) \sin (K[1])-\cos (K[1]) (K[1]+5))dK[1]-\sin (t) \int _1^t(\cos (K[2]) (2 K[2]+7)+3 \sin (K[2]))dK[2]-c_2 \sin (t)+c_1 (\cos (t)-\sin (t)) \\ y(t)\to 2 \sin (t) \int _1^t((K[1]+2) \sin (K[1])-\cos (K[1]) (K[1]+5))dK[1]+(\sin (t)+\cos (t)) \int _1^t(\cos (K[2]) (2 K[2]+7)+3 \sin (K[2]))dK[2]+2 c_1 \sin (t)+c_2 (\sin (t)+\cos (t)) \\ \end{align*}

✓ Sympy. Time used: 0.200 (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)
 

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