72.9.5 problem 5

Internal problem ID [14801]
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 : 5
Date solved : Tuesday, January 28, 2025 at 07:15:55 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 85

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

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

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 145

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

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