63.19.8 problem 3(b)

Internal problem ID [13222]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 202
Problem number : 3(b)
Date solved : Tuesday, January 28, 2025 at 05:12:38 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=-3 x+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=x+2 y \left (t \right )-1 \end{align*}

✓ Solution by Maple

Time used: 0.260 (sec). Leaf size: 87

dsolve([diff(x(t),t)=-3*x(t)+3*y(t),diff(y(t),t)=x(t)+2*y(t)-1],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {\left (-1+\sqrt {37}\right ) t}{2}} c_{2} +{\mathrm e}^{-\frac {\left (1+\sqrt {37}\right ) t}{2}} c_{1} +\frac {1}{3} \\ y &= \frac {{\mathrm e}^{\frac {\left (-1+\sqrt {37}\right ) t}{2}} c_{2} \sqrt {37}}{6}-\frac {{\mathrm e}^{-\frac {\left (1+\sqrt {37}\right ) t}{2}} c_{1} \sqrt {37}}{6}+\frac {5 \,{\mathrm e}^{\frac {\left (-1+\sqrt {37}\right ) t}{2}} c_{2}}{6}+\frac {5 \,{\mathrm e}^{-\frac {\left (1+\sqrt {37}\right ) t}{2}} c_{1}}{6}+\frac {1}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.445 (sec). Leaf size: 192

DSolve[{D[x[t],t]==-3*x[t]+3*y[t],D[y[t],t]==x[t]+2*y[t]-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{222} e^{-\frac {1}{2} \left (1+\sqrt {37}\right ) t} \left (74 e^{\frac {1}{2} \left (1+\sqrt {37}\right ) t}-3 \left (\left (5 \sqrt {37}-37\right ) c_1-6 \sqrt {37} c_2\right ) e^{\sqrt {37} t}+3 \left (\left (37+5 \sqrt {37}\right ) c_1-6 \sqrt {37} c_2\right )\right ) \\ y(t)\to \frac {1}{222} e^{-\frac {1}{2} \left (1+\sqrt {37}\right ) t} \left (74 e^{\frac {1}{2} \left (1+\sqrt {37}\right ) t}+3 \left (2 \sqrt {37} c_1+\left (37+5 \sqrt {37}\right ) c_2\right ) e^{\sqrt {37} t}-3 \left (2 \sqrt {37} c_1+\left (5 \sqrt {37}-37\right ) c_2\right )\right ) \\ \end{align*}