Internal problem ID [1863]
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: 11.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1} \relax (t )-x_{2} \relax (t )-t^{2}\\ x_{2}^{\prime }\relax (t )&=x_{1} \relax (t )+3 x_{2} \relax (t )+2 t \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 62
dsolve([diff(x__1(t),t)=1*x__1(t)-1*x__2(t)-t^2,diff(x__2(t),t)=1*x__1(t)+3*x__2(t)+2*t],[x__1(t), x__2(t)], singsol=all)
\[ x_{1} \relax (t ) = -c_{2} {\mathrm e}^{2 t}-{\mathrm e}^{2 t} t c_{1}+c_{1} {\mathrm e}^{2 t}+\frac {t}{2}+\frac {1}{8}+\frac {3 t^{2}}{4} \] \[ x_{2} \relax (t ) = c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{2 t} t c_{1}-\frac {t^{2}}{4}-t -\frac {3}{8} \]
✓ Solution by Mathematica
Time used: 0.187 (sec). Leaf size: 90
DSolve[{x1'[t]==1*x1[t]+3*x2[t]-t^2,x2'[t]==1*x1[t]+3*x2[t]+2*t},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{128} \left (-4 t \left (8 t^2+22 t+11\right )+32 c_1 \left (e^{4 t}+3\right )+96 c_2 \left (e^{4 t}-1\right )-11\right ) \\ \text {x2}(t)\to \frac {1}{384} \left (4 t \left (8 t^2+30 t-33\right )+96 (c_1+3 c_2) e^{4 t}-33-96 c_1+96 c_2\right ) \\ \end{align*}