64.16.14 problem 14

Internal problem ID [13617]
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 : Tuesday, January 28, 2025 at 05:53:57 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*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 41

dsolve([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],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.046 (sec). Leaf size: 158

DSolve[{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},{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*}