72.10.10 problem 10

Internal problem ID [14821]
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 3.2. page 277
Problem number : 10
Date solved : Tuesday, January 28, 2025 at 07:17:12 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 34

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} c_{1} +{\mathrm e}^{-2 t} c_{2} \\ y &= {\mathrm e}^{-3 t} c_{1} +\frac {{\mathrm e}^{-2 t} c_{2}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 58

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

\begin{align*} x(t)\to e^{-3 t} \left (c_1 \left (2 e^t-1\right )-2 c_2 \left (e^t-1\right )\right ) \\ y(t)\to e^{-3 t} \left (c_1 \left (e^t-1\right )-c_2 \left (e^t-2\right )\right ) \\ \end{align*}