57.2.3 problem 3

Internal problem ID [9057]
Book : First order enumerated odes
Section : section 2 (system of first order odes)
Problem number : 3
Date solved : Monday, January 27, 2025 at 05:31:43 PM
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 )&=y \left (t \right )+t +\sin \left (t \right )+\cos \left (t \right )\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+3 y \left (t \right )+{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.132 (sec). Leaf size: 44

dsolve([diff(x(t),t)+diff(y(t),t)-x(t)=y(t)+t+sin(t)+cos(t),diff(x(t),t)+diff(y(t),t)=2*x(t)+3*y(t)+exp(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 54

DSolve[{D[x[t],t]+D[y[t],t]-x[t]==y[t]+t+Sin[t]+Cos[t],D[x[t],t]+D[y[t],t]==2*x[t]+3*y[t]+Exp[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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