67.7.10 problem Problem 5(b)

Internal problem ID [14121]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 8.3 Systems of Linear Differential Equations (Variation of Parameters). Problems page 514
Problem number : Problem 5(b)
Date solved : Tuesday, January 28, 2025 at 06:14:48 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-2 y+24 \sin \left (t \right )\\ y^{\prime }&=9 x \left (t \right )-3 y+12 \cos \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = -1 \end{align*}

✓ Solution by Maple

Time used: 0.312 (sec). Leaf size: 43

dsolve([diff(x(t),t) = 3*x(t)-2*y(t)+24*sin(t), diff(y(t),t) = 9*x(t)-3*y(t)+12*cos(t), x(0) = 1, y(0) = -1], singsol=all)
 

\begin{align*} x \left (t \right ) &= -\frac {4 \sin \left (3 t \right )}{3}+\cos \left (3 t \right )+9 \sin \left (t \right ) \\ y &= \frac {7 \cos \left (3 t \right )}{2}-\frac {\sin \left (3 t \right )}{2}-\frac {9 \cos \left (t \right )}{2}+\frac {51 \sin \left (t \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 50

DSolve[{D[x[t],t]==3*x[t]-2*y[t]+24*Sin[t],D[y[t],t]==9*x[t]-3*y[t]+12*Cos[t]},{x[0]==1,y[0]==-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to 9 \sin (t)-\frac {4}{3} \sin (3 t)+\cos (3 t) \\ y(t)\to \frac {1}{2} (51 \sin (t)-\sin (3 t)-9 \cos (t)+7 \cos (3 t)) \\ \end{align*}