28.4.41 problem 7.41

Internal problem ID [4573]
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 : 7.41
Date solved : Tuesday, January 28, 2025 at 02:39:21 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-x_{1} \left (t \right )+2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+4 x_{2} \left (t \right )+\frac {{\mathrm e}^{3 t}}{1+{\mathrm e}^{2 t}} \end{align*}

Solution by Maple

Time used: 0.059 (sec). Leaf size: 70

dsolve([diff(x__1(t),t)=-x__1(t)+2*x__2(t),diff(x__2(t),t)=-3*x__1(t)+4*x__2(t)+exp(3*t)/(1+exp(2*t))],singsol=all)
 
\begin{align*} x_{1} \left (t \right ) &= 2 \arctan \left ({\mathrm e}^{t}\right ) {\mathrm e}^{2 t}+{\mathrm e}^{2 t} c_{1} -\ln \left (1+{\mathrm e}^{2 t}\right ) {\mathrm e}^{t}+c_{2} {\mathrm e}^{t} \\ x_{2} \left (t \right ) &= 3 \arctan \left ({\mathrm e}^{t}\right ) {\mathrm e}^{2 t}+\frac {3 \,{\mathrm e}^{2 t} c_{1}}{2}-\ln \left (1+{\mathrm e}^{2 t}\right ) {\mathrm e}^{t}+c_{2} {\mathrm e}^{t} \\ \end{align*}

Solution by Mathematica

Time used: 0.087 (sec). Leaf size: 100

DSolve[{D[x1[t],t]==-x1[t]+2*x2[t],D[x2[t],t]==-3*x1[t]+4*x2[t]+Exp[3*t]/(1+Exp[2*t])},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 
\begin{align*} \text {x1}(t)\to e^t \left (2 e^t \arctan \left (e^t\right )-\log \left (e^{2 t}+1\right )-2 c_1 e^t+2 c_2 e^t+3 c_1-2 c_2\right ) \\ \text {x2}(t)\to e^t \left (3 e^t \arctan \left (e^t\right )-\log \left (e^{2 t}+1\right )-3 c_1 e^t+3 c_2 e^t+3 c_1-2 c_2\right ) \\ \end{align*}