75.32.5 problem 829

Internal problem ID [17282]
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 23.3 dAlemberts method. Exercises page 243
Problem number : 829
Date solved : Tuesday, January 28, 2025 at 09:58:51 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+4 y \left (t \right )+\cos \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )-2 y \left (t \right )+\sin \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.057 (sec). Leaf size: 35

dsolve([diff(x(t),t)=2*x(t)+4*y(t)+cos(t),diff(y(t),t)=-x(t)-2*y(t)+sin(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= -2 \cos \left (t \right )-3 \sin \left (t \right )+c_{1} t +c_{2} \\ y \left (t \right ) &= 2 \sin \left (t \right )+\frac {c_{1}}{4}-\frac {c_{1} t}{2}-\frac {c_{2}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 150

DSolve[{D[x[t],t]==2*x[t]+4*y[t]+Cos[t],D[y[t],t]==-x[t]-2*y[t]+Sin[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to (2 t+1) \int _1^t(\cos (K[1]) (1-2 K[1])-4 K[1] \sin (K[1]))dK[1]+4 t \int _1^t(\cos (K[2]) K[2]+2 \sin (K[2]) K[2]+\sin (K[2]))dK[2]+2 c_1 t+4 c_2 t+c_1 \\ y(t)\to -t \int _1^t(\cos (K[1]) (1-2 K[1])-4 K[1] \sin (K[1]))dK[1]+(1-2 t) \int _1^t(\cos (K[2]) K[2]+2 \sin (K[2]) K[2]+\sin (K[2]))dK[2]+c_1 (-t)-2 c_2 t+c_2 \\ \end{align*}