72.9.9 problem 9

Internal problem ID [14805]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.1. page 258
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 07:16:59 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=\beta y\\ y^{\prime }&=\gamma x \left (t \right )-y \end{align*}

✓ Solution by Maple

Time used: 0.125 (sec). Leaf size: 118

dsolve([diff(x(t),t)=beta*y(t),diff(y(t),t)=gamma*x(t)-y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {4 \beta \gamma +1}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {4 \beta \gamma +1}\right ) t}{2}} \\ y &= \frac {\left (-\frac {1}{2}+\frac {\sqrt {4 \beta \gamma +1}}{2}\right ) c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {4 \beta \gamma +1}\right ) t}{2}}}{\beta }+\frac {\left (-\frac {{\mathrm e}^{-\frac {\left (1+\sqrt {4 \beta \gamma +1}\right ) t}{2}} \sqrt {4 \beta \gamma +1}}{2}-\frac {{\mathrm e}^{-\frac {\left (1+\sqrt {4 \beta \gamma +1}\right ) t}{2}}}{2}\right ) c_{2}}{\beta } \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 202

DSolve[{D[x[t],t]==\[Beta]*y[t],D[y[t],t]==\[Gamma]*x[t]-y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {e^{-\frac {1}{2} t \left (\sqrt {4 \beta \gamma +1}+1\right )} \left (c_1 \left (\sqrt {4 \beta \gamma +1}+\left (\sqrt {4 \beta \gamma +1}+1\right ) e^{t \sqrt {4 \beta \gamma +1}}-1\right )+2 \beta c_2 \left (e^{t \sqrt {4 \beta \gamma +1}}-1\right )\right )}{2 \sqrt {4 \beta \gamma +1}} \\ y(t)\to \frac {e^{-\frac {1}{2} t \left (\sqrt {4 \beta \gamma +1}+1\right )} \left (2 \gamma c_1 \left (e^{t \sqrt {4 \beta \gamma +1}}-1\right )+c_2 \left (\sqrt {4 \beta \gamma +1}+\left (\sqrt {4 \beta \gamma +1}-1\right ) e^{t \sqrt {4 \beta \gamma +1}}+1\right )\right )}{2 \sqrt {4 \beta \gamma +1}} \\ \end{align*}