14.25.1 problem Example 1, page 361

Internal problem ID [2758]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.12, Systems of differential equations. The nonhomogeneous equation. variation of parameters. Page 366
Problem number : Example 1, page 361
Date solved : Monday, January 27, 2025 at 06:12:49 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right )+{\mathrm e}^{t} \cos \left (2 t \right ) \end{align*}

✓ Solution by Maple

Time used: 1.038 (sec). Leaf size: 90

dsolve([diff(x__1(t),t)=1*x__1(t)+0*x__2(t)+0*x__3(t),diff(x__2(t),t)=2*x__1(t)+1*x__2(t)-2*x__3(t),diff(x__3(t),t)=3*x__1(t)+2*x__2(t)+1*x__3(t)+exp(t)*cos(2*t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_3 \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-3 c_3 -3 c_3 \cos \left (2 t \right )+2 c_2 \sin \left (2 t \right )-t \sin \left (2 t \right )+2 c_1 \cos \left (2 t \right )\right )}{2} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (4 c_1 \sin \left (2 t \right )-6 c_3 \sin \left (2 t \right )-4 c_2 \cos \left (2 t \right )+2 t \cos \left (2 t \right )+\sin \left (2 t \right )+4 c_3 \right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 103

DSolve[{D[ x1[t],t]==1*x1[t]+0*x2[t]+0*x3[t],D[ x2[t],t]==2*x1[t]+1*x2[t]-2*x3[t],D[ x3[t],t]==3*x1[t]+2*x2[t]+1*x3[t]+Exp[t]*Cos[2*t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 e^t \\ \text {x2}(t)\to -\frac {1}{8} e^t ((1-12 c_1-8 c_2) \cos (2 t)+4 (t-2 c_1+2 c_3) \sin (2 t)+12 c_1) \\ \text {x3}(t)\to \frac {1}{8} e^t (4 (t-2 c_1+2 c_3) \cos (2 t)+(1+12 c_1+8 c_2) \sin (2 t)+8 c_1) \\ \end{align*}