72.12.6 problem 6

Internal problem ID [14853]
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 3.5 page 327
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 07:17:42 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 24

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (1-t \right ) \\ y &= -t \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 26

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

\begin{align*} x(t)\to -e^{3 t} (t-1) \\ y(t)\to -e^{3 t} t \\ \end{align*}