Internal problem ID [13103]
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.
ODE order: 1.
ODE degree: 1.
Solve \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 \[ [x \left (0\right ) = 1, y \left (0\right ) = -1] \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 61
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 \left (t \right ) &= -\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: 0.03 (sec). Leaf size: 94
DSolve[{x'[t]==1*x[t]+4*y[t],y'[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*}