72.11.13 problem 24

Internal problem ID [14846]
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 : 24
Date solved : Tuesday, January 28, 2025 at 07:17:36 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-\frac {9 x \left (t \right )}{10}-2 y\\ y^{\prime }&=x \left (t \right )+\frac {11 y}{10} \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 34

dsolve([diff(x(t),t) = -9/10*x(t)-2*y(t), diff(y(t),t) = x(t)+11/10*y(t), x(0) = 1, y(0) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{10}} \left (-3 \sin \left (t \right )+\cos \left (t \right )\right ) \\ y &= -\frac {{\mathrm e}^{\frac {t}{10}} \left (-4 \sin \left (t \right )-2 \cos \left (t \right )\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 38

DSolve[{D[x[t],t]==-9/10*x[t]-2*y[t],D[y[t],t]==x[t]+11/10*y[t]},{x[0]==1,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{t/10} (\cos (t)-3 \sin (t)) \\ y(t)\to e^{t/10} (2 \sin (t)+\cos (t)) \\ \end{align*}