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 }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )-t^{2}\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+3 x_{2} \left (t \right )+2 t \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 63
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],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{2 t} t c_{1} +\frac {3 t^{2}}{4}+\frac {t}{2}+\frac {1}{8} \\ x_{2} \left (t \right ) &= -\frac {t^{2}}{4}-c_{2} {\mathrm e}^{2 t}-{\mathrm e}^{2 t} t c_{1} -t -\frac {3}{8}-c_{1} {\mathrm e}^{2 t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.27 (sec). Leaf size: 94
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 (-32 t^3-88 t^2-44 t+32 c_1 \left (e^{4 t}+3\right )+96 c_2 e^{4 t}-11-96 c_2\right ) \\ \text {x2}(t)\to \frac {1}{384} \left (32 t^3+120 t^2-132 t+96 c_1 \left (e^{4 t}-1\right )+288 c_2 e^{4 t}-33+96 c_2\right ) \\ \end{align*}