Internal problem ID [1854]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1} \relax (t )\\ x_{2}^{\prime }\relax (t )&=2 x_{1} \relax (t )+x_{2} \relax (t )-2 x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=3 x_{1} \relax (t )+2 x_{2} \relax (t )+x_{3} \relax (t )+{\mathrm e}^{t} \cos \left (2 t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.437 (sec). Leaf size: 85
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)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = c_{1} {\mathrm e}^{t} \] \[ x_{2} \relax (t ) = \frac {{\mathrm e}^{t} \left (2 c_{3} \cos \left (2 t \right )-2 c_{2} \sin \left (2 t \right )-t \sin \left (2 t \right )-\cos \left (2 t \right )-3 c_{1}\right )}{2} \] \[ x_{3} \relax (t ) = \frac {{\mathrm e}^{t} \left (4 c_{2} \cos \left (2 t \right )+2 t \cos \left (2 t \right )+4 \sin \left (2 t \right ) c_{3}-\sin \left (2 t \right )+4 c_{1}\right )}{4} \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 103
DSolve[{x1'[t]==1*x1[t]+0*x2[t]+0*x3[t],x2'[t]==2*x1[t]+1*x2[t]-2*x3[t],x3'[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*}