69.1.121 problem 172

Internal problem ID [14274]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 172
Date solved : Tuesday, January 28, 2025 at 06:25:15 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.334 (sec). Leaf size: 36

dsolve([4*diff(x(t),t)-diff(y(t),t)+3*x(t)=sin(t),diff(x(t),t)+y(t)=cos(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.165 (sec). Leaf size: 289

DSolve[{4*D[x[t],t]-D[y[t],t]+3*x[t]==Sin[t],D[x[t],t]+y[t]==Cos[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-3 t} \left (\left (3 e^{2 t}-1\right ) \int _1^t\frac {1}{2} e^{K[1]} \left (\left (-1+3 e^{2 K[1]}\right ) \cos (K[1])-\left (-1+e^{2 K[1]}\right ) \sin (K[1])\right )dK[1]-\left (e^{2 t}-1\right ) \int _1^t\frac {1}{2} e^{K[2]} \left (\left (-1+9 e^{2 K[2]}\right ) \cos (K[2])+\left (1-3 e^{2 K[2]}\right ) \sin (K[2])\right )dK[2]+3 c_1 e^{2 t}-c_2 e^{2 t}-c_1+c_2\right ) \\ y(t)\to \frac {1}{2} e^{-3 t} \left (3 \left (e^{2 t}-1\right ) \int _1^t\frac {1}{2} e^{K[1]} \left (\left (-1+3 e^{2 K[1]}\right ) \cos (K[1])-\left (-1+e^{2 K[1]}\right ) \sin (K[1])\right )dK[1]-\left (e^{2 t}-3\right ) \int _1^t\frac {1}{2} e^{K[2]} \left (\left (-1+9 e^{2 K[2]}\right ) \cos (K[2])+\left (1-3 e^{2 K[2]}\right ) \sin (K[2])\right )dK[2]+3 c_1 e^{2 t}-c_2 e^{2 t}-3 c_1+3 c_2\right ) \\ \end{align*}