28.4.45 problem 7.45

Internal problem ID [4577]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.45
Date solved : Monday, January 27, 2025 at 09:23:43 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 )+27 t\\ x_{2}^{\prime }\left (t \right )&=-x_{1} \left (t \right )+4 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 49

dsolve([diff(x__1(t),t)=2*x__1(t)+x__2(t)+27*t,diff(x__2(t),t)=-x__1(t)+4*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{3 t}+{\mathrm e}^{3 t} t c_{1} -12 t -5 \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{3 t}+{\mathrm e}^{3 t} t c_{1} +c_{1} {\mathrm e}^{3 t}-2-3 t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.059 (sec). Leaf size: 62

DSolve[{D[x1[t],t]==2*x1[t]+x2[t]+27*t,D[x2[t],t]==-x1[t]+4*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 e^{3 t}+t \left (-12+(c_2-c_1) e^{3 t}\right )-5 \\ \text {x2}(t)\to c_2 e^{3 t}+t \left (-3+(c_2-c_1) e^{3 t}\right )-2 \\ \end{align*}