72.11.5 problem 7

Internal problem ID [14838]
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 : 7
Date solved : Tuesday, January 28, 2025 at 07:17:27 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-6 y\\ y^{\prime }&=2 x \left (t \right )+y \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 62

dsolve([diff(x(t),t) = 2*x(t)-6*y(t), diff(y(t),t) = 2*x(t)+y(t), x(0) = 2, y(0) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (-\frac {10 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+2 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right ) \\ y &= \frac {{\mathrm e}^{\frac {3 t}{2}} \left (\frac {84 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+12 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right )}{12} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 94

DSolve[{D[x[t],t]==2*x[t]-6*y[t],D[y[t],t]==2*x[t]+y[t]},{x[0]==2,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {2}{47} e^{3 t/2} \left (47 \cos \left (\frac {\sqrt {47} t}{2}\right )-5 \sqrt {47} \sin \left (\frac {\sqrt {47} t}{2}\right )\right ) \\ y(t)\to \frac {1}{47} e^{3 t/2} \left (7 \sqrt {47} \sin \left (\frac {\sqrt {47} t}{2}\right )+47 \cos \left (\frac {\sqrt {47} t}{2}\right )\right ) \\ \end{align*}