72.9.6 problem 6

Internal problem ID [14802]
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 : 6
Date solved : Tuesday, January 28, 2025 at 07:15:56 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 119

dsolve([diff(x(t),t)=3*y(t),diff(y(t),t)=3*Pi*y(t)-1/3*x(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (3 \pi -\sqrt {9 \pi ^{2}-4}\right ) t}{2}}+c_{2} {\mathrm e}^{\frac {\left (3 \pi +\sqrt {9 \pi ^{2}-4}\right ) t}{2}} \\ y &= \left (\frac {\pi }{2}+\frac {\sqrt {9 \pi ^{2}-4}}{6}\right ) c_{2} {\mathrm e}^{\frac {\left (3 \pi +\sqrt {9 \pi ^{2}-4}\right ) t}{2}}+\left (\frac {\pi }{2}-\frac {\sqrt {9 \pi ^{2}-4}}{6}\right ) c_{1} {\mathrm e}^{\frac {\left (3 \pi -\sqrt {9 \pi ^{2}-4}\right ) t}{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 233

DSolve[{D[x[t],t]==3*y[t],D[y[t],t]==3*Pi*y[t]-1/3*x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {9 \pi ^2-4}-3 \pi \right ) t} \left (\sqrt {9 \pi ^2-4} c_1 \left (e^{\sqrt {9 \pi ^2-4} t}+1\right )-3 \pi c_1 \left (e^{\sqrt {9 \pi ^2-4} t}-1\right )+6 c_2 \left (e^{\sqrt {9 \pi ^2-4} t}-1\right )\right )}{2 \sqrt {9 \pi ^2-4}} \\ y(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {9 \pi ^2-4}-3 \pi \right ) t} \left (3 c_2 \left (3 \pi \left (e^{\sqrt {9 \pi ^2-4} t}-1\right )+\sqrt {9 \pi ^2-4} \left (e^{\sqrt {9 \pi ^2-4} t}+1\right )\right )-2 c_1 \left (e^{\sqrt {9 \pi ^2-4} t}-1\right )\right )}{6 \sqrt {9 \pi ^2-4}} \\ \end{align*}