72.11.4 problem 6

Internal problem ID [14837]
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 : 6
Date solved : Tuesday, January 28, 2025 at 07:17:26 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 62

dsolve([diff(x(t),t) = 2*y(t), diff(y(t),t) = -2*x(t)-y(t), x(0) = -1, y(0) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (\frac {\sqrt {15}\, \sin \left (\frac {\sqrt {15}\, t}{2}\right )}{5}-\cos \left (\frac {\sqrt {15}\, t}{2}\right )\right ) \\ y &= -\frac {{\mathrm e}^{-\frac {t}{2}} \left (-\frac {4 \sqrt {15}\, \sin \left (\frac {\sqrt {15}\, t}{2}\right )}{5}-4 \cos \left (\frac {\sqrt {15}\, t}{2}\right )\right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 92

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

\begin{align*} x(t)\to \frac {1}{5} e^{-t/2} \left (\sqrt {15} \sin \left (\frac {\sqrt {15} t}{2}\right )-5 \cos \left (\frac {\sqrt {15} t}{2}\right )\right ) \\ y(t)\to \frac {1}{5} e^{-t/2} \left (\sqrt {15} \sin \left (\frac {\sqrt {15} t}{2}\right )+5 \cos \left (\frac {\sqrt {15} t}{2}\right )\right ) \\ \end{align*}