75.31.8 problem 822

Internal problem ID [17275]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 23.2 The method of undetermined coefficients. Exercises page 239
Problem number : 822
Date solved : Tuesday, January 28, 2025 at 09:58:45 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.079 (sec). Leaf size: 38

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

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

✓ Solution by Mathematica

Time used: 0.168 (sec). Leaf size: 97

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

\begin{align*} x(t)\to \int _1^te^{-K[1]} \left (2 K[1]-e^{K[1]} (K[1]-1) \sin (K[1])-1\right )dK[1]-t \int _1^t\left (2 e^{-K[2]}-\sin (K[2])\right )dK[2]-c_2 t+c_1 \\ y(t)\to \int _1^t\left (2 e^{-K[2]}-\sin (K[2])\right )dK[2]+c_2 \\ \end{align*}