67.7.14 problem Problem 6(b)

Internal problem ID [14125]
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 6(b)
Date solved : Tuesday, January 28, 2025 at 06:15:17 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y-z \left (t \right )+5 \sin \left (t \right )\\ y^{\prime }&=y+z \left (t \right )-10 \cos \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+z \left (t \right )+2 \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 2\\ z \left (0\right ) = 3 \end{align*}

✓ Solution by Maple

Time used: 0.834 (sec). Leaf size: 70

dsolve([diff(x(t),t) = 2*x(t)+y(t)-z(t)+5*sin(t), diff(y(t),t) = y(t)+z(t)-10*cos(t), diff(z(t),t) = x(t)+z(t)+2, x(0) = 1, y(0) = 2, z(0) = 3], singsol=all)
 

\begin{align*} x \left (t \right ) &= -2 \cos \left (t \right )-1-3 \,{\mathrm e}^{t} \sin \left (t \right )+4 \,{\mathrm e}^{t} \cos \left (t \right ) \\ y &= -4 \sin \left (t \right )+5 \cos \left (t \right )+1+3 \,{\mathrm e}^{t} \sin \left (t \right )-4 \,{\mathrm e}^{t} \cos \left (t \right ) \\ z \left (t \right ) &= -1-\sin \left (t \right )+\cos \left (t \right )+3 \,{\mathrm e}^{t} \cos \left (t \right )+4 \,{\mathrm e}^{t} \sin \left (t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.098 (sec). Leaf size: 2170

DSolve[{D[x[t],t]==2*x[t]+y[t]-z[t]+5*Sin[t],D[y[t],t]==y[t]+z[t]-10*Cos[t],D[z[t],t]==x[t]+z[t]+2},{x[0]==1,y[0]==2,z[0]==3},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

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