67.7.15 problem Problem 6(c)

Internal problem ID [14126]
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(c)
Date solved : Tuesday, January 28, 2025 at 06:15:19 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+3 y+z \left (t \right )+5 \sin \left (2 t \right )\\ y^{\prime }&=x \left (t \right )-5 y-3 z \left (t \right )+5 \cos \left (2 t \right )\\ z^{\prime }\left (t \right )&=-3 x \left (t \right )+7 y+3 z \left (t \right )+23 \,{\mathrm e}^{t} \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: 3.328 (sec). Leaf size: 131

dsolve([diff(x(t),t) = -3*x(t)+3*y(t)+z(t)+5*sin(2*t), diff(y(t),t) = x(t)-5*y(t)-3*z(t)+5*cos(2*t), diff(z(t),t) = -3*x(t)+7*y(t)+3*z(t)+23*exp(t), x(0) = 1, y(0) = 2, z(0) = 3], singsol=all)
 

\begin{align*} x \left (t \right ) &= -\frac {69 \,{\mathrm e}^{t}}{26}+\frac {\cos \left (2 t \right )}{2}+\sin \left (2 t \right )+\frac {21 \,{\mathrm e}^{-t}}{2}-\frac {191 \cos \left (2 t \right ) {\mathrm e}^{-2 t}}{26}+\frac {16 \sin \left (2 t \right ) {\mathrm e}^{-2 t}}{13} \\ y &= -\frac {253 \,{\mathrm e}^{t}}{26}-\frac {5 \sin \left (2 t \right )}{2}+\frac {21 \,{\mathrm e}^{-t}}{2}+\frac {191 \sin \left (2 t \right ) {\mathrm e}^{-2 t}}{26}+\frac {16 \cos \left (2 t \right ) {\mathrm e}^{-2 t}}{13} \\ z \left (t \right ) &= \frac {483 \,{\mathrm e}^{t}}{26}+\frac {9 \sin \left (2 t \right )}{2}+\frac {7 \cos \left (2 t \right )}{2}-\frac {21 \,{\mathrm e}^{-t}}{2}-\frac {223 \cos \left (2 t \right ) {\mathrm e}^{-2 t}}{26}-\frac {159 \sin \left (2 t \right ) {\mathrm e}^{-2 t}}{26} \\ \end{align*}

✓ Solution by Mathematica

Time used: 7.771 (sec). Leaf size: 3703

DSolve[{D[x[t],t]==-3*x[t]+3*y[t]+z[t]+5*Sin[3*t],D[y[t],t]==x[t]-5*y[t]-3*z[t]+5*Cos[2*t],D[z[t],t]==-3*x[t]+7*y[t]+3*z[t]+23*Exp[t]},{x[0]==1,y[0]==2,z[0]==3},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

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