72.11.2 problem 4

Internal problem ID [14835]
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 : 4
Date solved : Tuesday, January 28, 2025 at 07:17:24 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+2 y\\ y^{\prime }&=-4 x \left (t \right )+6 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: 32

dsolve([diff(x(t),t) = 2*x(t)+2*y(t), diff(y(t),t) = -4*x(t)+6*y(t), x(0) = 1, y(0) = 1], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 35

DSolve[{D[x[t],t]==2*x[t]+2*y[t],D[y[t],t]==-4*x[t]+6*y[t]},{x[0]==1,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{4 t} \cos (2 t) \\ y(t)\to e^{4 t} (\cos (2 t)-\sin (2 t)) \\ \end{align*}