72.11.3 problem 5

Internal problem ID [14836]
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 3.4 page 310
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 07:17:25 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )-5 y\\ y^{\prime }&=3 x \left (t \right )+y \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 4\\ y \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 47

dsolve([diff(x(t),t) = -3*x(t)-5*y(t), diff(y(t),t) = 3*x(t)+y(t), x(0) = 4, y(0) = 0], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (-\frac {8 \sqrt {11}\, \sin \left (\sqrt {11}\, t \right )}{11}+4 \cos \left (\sqrt {11}\, t \right )\right ) \\ y &= \frac {12 \,{\mathrm e}^{-t} \sqrt {11}\, \sin \left (\sqrt {11}\, t \right )}{11} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 63

DSolve[{D[x[t],t]==-3*x[t]-5*y[t],D[y[t],t]==3*x[t]+y[t]},{x[0]==4,y[0]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {4}{11} e^{-t} \left (11 \cos \left (\sqrt {11} t\right )-2 \sqrt {11} \sin \left (\sqrt {11} t\right )\right ) \\ y(t)\to \frac {12 e^{-t} \sin \left (\sqrt {11} t\right )}{\sqrt {11}} \\ \end{align*}