72.9.8 problem 8

Internal problem ID [14804]
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 : 8
Date solved : Tuesday, January 28, 2025 at 07:16:58 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+2 \pi y\\ y^{\prime }&=4 x \left (t \right )-y \end{align*}

✓ Solution by Maple

Time used: 0.057 (sec). Leaf size: 118

dsolve([diff(x(t),t)=-3*x(t)+2*Pi*y(t),diff(y(t),t)=4*x(t)-y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-\left (2+\sqrt {1+8 \pi }\right ) t}+c_{2} {\mathrm e}^{\left (-2+\sqrt {1+8 \pi }\right ) t} \\ y &= -\frac {c_{1} {\mathrm e}^{-\left (2+\sqrt {1+8 \pi }\right ) t} \sqrt {1+8 \pi }-c_{2} {\mathrm e}^{\left (-2+\sqrt {1+8 \pi }\right ) t} \sqrt {1+8 \pi }-c_{1} {\mathrm e}^{-\left (2+\sqrt {1+8 \pi }\right ) t}-c_{2} {\mathrm e}^{\left (-2+\sqrt {1+8 \pi }\right ) t}}{2 \pi } \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 189

DSolve[{D[x[t],t]==-3*x[t]+2*Pi*y[t],D[y[t],t]==4*x[t]-y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {e^{-\left (\left (2+\sqrt {1+8 \pi }\right ) t\right )} \left (c_1 \left (\left (\sqrt {1+8 \pi }-1\right ) e^{2 \sqrt {1+8 \pi } t}+1+\sqrt {1+8 \pi }\right )+2 \pi c_2 \left (e^{2 \sqrt {1+8 \pi } t}-1\right )\right )}{2 \sqrt {1+8 \pi }} \\ y(t)\to \frac {e^{-\left (\left (2+\sqrt {1+8 \pi }\right ) t\right )} \left (4 c_1 \left (e^{2 \sqrt {1+8 \pi } t}-1\right )+c_2 \left (\left (1+\sqrt {1+8 \pi }\right ) e^{2 \sqrt {1+8 \pi } t}-1+\sqrt {1+8 \pi }\right )\right )}{2 \sqrt {1+8 \pi }} \\ \end{align*}