67.5.15 problem Problem 3(d)

Internal problem ID [14101]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6. Introduction to Systems of ODEs. Problems page 408
Problem number : Problem 3(d)
Date solved : Tuesday, January 28, 2025 at 06:14:27 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )+5 x \left (t \right )-2 y&=0\\ y^{\prime }+2 x \left (t \right )-y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 82

dsolve([diff(x(t),t)+5*x(t)-2*y(t)=0,diff(y(t),t)+2*x(t)-y(t)=0],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 148

DSolve[{D[x[t],t]+5*x[t]-2*y[t]==0,D[y[t],t]+2*x[t]-y[t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{10} e^{-\left (\left (2+\sqrt {5}\right ) t\right )} \left (c_1 \left (\left (5-3 \sqrt {5}\right ) e^{2 \sqrt {5} t}+5+3 \sqrt {5}\right )+2 \sqrt {5} c_2 \left (e^{2 \sqrt {5} t}-1\right )\right ) \\ y(t)\to \frac {1}{10} e^{-\left (\left (2+\sqrt {5}\right ) t\right )} \left (c_2 \left (\left (5+3 \sqrt {5}\right ) e^{2 \sqrt {5} t}+5-3 \sqrt {5}\right )-2 \sqrt {5} c_1 \left (e^{2 \sqrt {5} t}-1\right )\right ) \\ \end{align*}