67.6.7 problem Problem 4(g)

Internal problem ID [14111]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6.4 Reduction to a single ODE. Problems page 415
Problem number : Problem 4(g)
Date solved : Tuesday, January 28, 2025 at 06:14:36 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.469 (sec). Leaf size: 50

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

\begin{align*} x \left (t \right ) &= -8 \,{\mathrm e}^{-\frac {t}{8}} c_{1} -\frac {6 \sin \left (t \right )}{65}-\frac {17 \cos \left (t \right )}{65}+2 t +c_{2} \\ y &= 12 \,{\mathrm e}^{-\frac {t}{8}} c_{1} -\frac {7 \cos \left (t \right )}{65}+\frac {9 \sin \left (t \right )}{65}+8-3 t -c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.144 (sec). Leaf size: 266

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

\begin{align*} x(t)\to \left (2 e^{t/4}-1\right ) \int _1^t\frac {1}{2} e^{-\frac {K[1]}{4}} \left (K[1]+\left (3-2 e^{\frac {K[1]}{4}}\right ) \sin (K[1])\right )dK[1]+2 \left (e^{t/4}-1\right ) \int _1^t\frac {1}{4} e^{-\frac {K[2]}{4}} \left (\left (-3+4 e^{\frac {K[2]}{4}}\right ) \sin (K[2])-K[2]\right )dK[2]+c_1 \left (2 e^{t/4}-1\right )+2 c_2 \left (e^{t/4}-1\right ) \\ y(t)\to -\left (e^{t/4}-1\right ) \int _1^t\frac {1}{2} e^{-\frac {K[1]}{4}} \left (K[1]+\left (3-2 e^{\frac {K[1]}{4}}\right ) \sin (K[1])\right )dK[1]-\left (e^{t/4}-2\right ) \int _1^t\frac {1}{4} e^{-\frac {K[2]}{4}} \left (\left (-3+4 e^{\frac {K[2]}{4}}\right ) \sin (K[2])-K[2]\right )dK[2]+c_1 \left (-e^{t/4}\right )-c_2 e^{t/4}+c_1+2 c_2 \\ \end{align*}