50.29.7 problem 3(c)

Internal problem ID [8203]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page 400
Problem number : 3(c)
Date solved : Monday, January 27, 2025 at 03:45:46 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-5 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )+2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 85

dsolve([diff(x(t),t)=3*x(t)-5*y(t),diff(y(t),t)=-x(t)+2*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}} \\ y &= -\frac {c_{1} {\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{10}+\frac {c_{2} {\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{10}+\frac {c_{1} {\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}}}{10}+\frac {c_{2} {\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}}}{10} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 144

DSolve[{D[x[t],t]==3*x[t]-5*y[t],D[y[t],t]==-x[t]+2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{42} e^{-\frac {1}{2} \left (\sqrt {21}-5\right ) t} \left (c_1 \left (\left (21+\sqrt {21}\right ) e^{\sqrt {21} t}+21-\sqrt {21}\right )-10 \sqrt {21} c_2 \left (e^{\sqrt {21} t}-1\right )\right ) \\ y(t)\to -\frac {1}{42} e^{-\frac {1}{2} \left (\sqrt {21}-5\right ) t} \left (2 \sqrt {21} c_1 \left (e^{\sqrt {21} t}-1\right )+c_2 \left (\left (\sqrt {21}-21\right ) e^{\sqrt {21} t}-21-\sqrt {21}\right )\right ) \\ \end{align*}