14.25.10 problem 11

Internal problem ID [2767]
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
Date solved : Monday, January 27, 2025 at 06:13:01 AM
CAS classification : system_of_ODEs

\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: 0.026 (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],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 -{\mathrm e}^{2 t} c_1 -t -\frac {3}{8} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.331 (sec). Leaf size: 94

DSolve[{D[ x1[t],t]==1*x1[t]+3*x2[t]-t^2,D[ x2[t],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*}