14.25.9 problem 10

Internal problem ID [2766]
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 : 10
Date solved : Monday, January 27, 2025 at 06:13:00 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 81

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\sqrt {7}\, t} c_2 +{\mathrm e}^{-\sqrt {7}\, t} c_1 +3 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= \sqrt {7}\, {\mathrm e}^{\sqrt {7}\, t} c_2 -\sqrt {7}\, {\mathrm e}^{-\sqrt {7}\, t} c_1 +2 \,{\mathrm e}^{3 t}-2 \,{\mathrm e}^{\sqrt {7}\, t} c_2 -2 \,{\mathrm e}^{-\sqrt {7}\, t} c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.599 (sec). Leaf size: 171

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

\begin{align*} \text {x1}(t)\to \frac {1}{14} e^{-\sqrt {7} t} \left (42 e^{\left (3+\sqrt {7}\right ) t}+\left (\left (7+2 \sqrt {7}\right ) c_1+\sqrt {7} c_2\right ) e^{2 \sqrt {7} t}+\left (7-2 \sqrt {7}\right ) c_1-\sqrt {7} c_2\right ) \\ \text {x2}(t)\to \frac {1}{14} e^{-\sqrt {7} t} \left (28 e^{\left (3+\sqrt {7}\right ) t}+\left (3 \sqrt {7} c_1+\left (7-2 \sqrt {7}\right ) c_2\right ) e^{2 \sqrt {7} t}-3 \sqrt {7} c_1+\left (7+2 \sqrt {7}\right ) c_2\right ) \\ \end{align*}