7.25.16 problem 16

Internal problem ID [636]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 16
Date solved : Monday, January 27, 2025 at 02:56:34 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-50 x_{1} \left (t \right )+20 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=100 x_{1} \left (t \right )-60 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 35

dsolve([diff(x__1(t),t)=-50*x__1(t)+20*x__2(t),diff(x__2(t),t)=100*x__1(t)-60*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-100 t}+c_2 \,{\mathrm e}^{-10 t} \\ x_{2} \left (t \right ) &= -\frac {5 c_1 \,{\mathrm e}^{-100 t}}{2}+2 c_2 \,{\mathrm e}^{-10 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 74

DSolve[{D[x1[t],t]==-50*x1[t]+20*x2[t],D[x2[t],t]==100*x1[t]-60*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{9} e^{-100 t} \left (c_1 \left (5 e^{90 t}+4\right )+2 c_2 \left (e^{90 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-100 t} \left (10 c_1 \left (e^{90 t}-1\right )+c_2 \left (4 e^{90 t}+5\right )\right ) \\ \end{align*}