9.4.15 problem problem 15

Internal problem ID [979]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 15
Date solved : Wednesday, February 05, 2025 at 04:51:26 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=7 x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )+3 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 60

dsolve([diff(x__1(t),t)=7*x__1(t)-5*x__2(t),diff(x__2(t),t)=4*x__1(t)+3*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{5 t} \left (c_1 \sin \left (4 t \right )+c_2 \cos \left (4 t \right )\right ) \\ x_{2} \left (t \right ) &= -\frac {2 \,{\mathrm e}^{5 t} \left (2 c_1 \cos \left (4 t \right )-c_2 \cos \left (4 t \right )-c_1 \sin \left (4 t \right )-2 c_2 \sin \left (4 t \right )\right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 72

DSolve[{D[ x1[t],t]==7*x1[t]-5*x2[t],D[ x2[t],t]==4*x1[t]+3*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{5 t} (4 c_1 \cos (4 t)+(2 c_1-5 c_2) \sin (4 t)) \\ \text {x2}(t)\to \frac {1}{2} e^{5 t} (2 c_2 \cos (4 t)+(2 c_1-c_2) \sin (4 t)) \\ \end{align*}