72.11.6 problem 8

Internal problem ID [14839]
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 : 8
Date solved : Tuesday, January 28, 2025 at 07:17:28 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+4 y\\ y^{\prime }&=-3 x \left (t \right )+2 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: 0.046 (sec). Leaf size: 60

dsolve([diff(x(t),t) = x(t)+4*y(t), diff(y(t),t) = -3*x(t)+2*y(t), x(0) = 1, y(0) = -1], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (-\frac {9 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+\cos \left (\frac {\sqrt {47}\, t}{2}\right )\right ) \\ y &= -\frac {{\mathrm e}^{\frac {3 t}{2}} \left (\frac {56 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+8 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right )}{8} \\ \end{align*}

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]==1*x[t]+4*y[t],D[y[t],t]==-3*x[t]+2*y[t]},{x[0]==1,y[0]==-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{47} e^{3 t/2} \left (47 \cos \left (\frac {\sqrt {47} t}{2}\right )-9 \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*}