75.29.4 problem 805

Internal problem ID [17258]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 22. Integration of homogeneous linear systems with constant coefficients. Eulers method. Exercises page 230
Problem number : 805
Date solved : Tuesday, January 28, 2025 at 09:58:31 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 29

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

\begin{align*} x \left (t \right ) &= -{\mathrm e}^{3 t}+{\mathrm e}^{2 t} \\ y \left (t \right ) &= -2 \,{\mathrm e}^{3 t}+{\mathrm e}^{2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 33

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

\begin{align*} x(t)\to -e^{2 t} \left (e^t-1\right ) \\ y(t)\to e^{2 t}-2 e^{3 t} \\ \end{align*}