72.10.2 problem 2

Internal problem ID [14813]
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 : 2
Date solved : Tuesday, January 28, 2025 at 07:17:06 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )-2 y\\ y^{\prime }&=-x \left (t \right )-3 y \end{align*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 35

dsolve([diff(x(t),t)=-4*x(t)-2*y(t),diff(y(t),t)=-x(t)-3*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-5 t} c_{1} +{\mathrm e}^{-2 t} c_{2} \\ y &= \frac {{\mathrm e}^{-5 t} c_{1}}{2}-{\mathrm e}^{-2 t} c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 71

DSolve[{D[x[t],t]==-4*x[t]-2*y[t],D[y[t],t]==-x[t]-3*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{-5 t} \left (c_1 \left (e^{3 t}+2\right )-2 c_2 \left (e^{3 t}-1\right )\right ) \\ y(t)\to \frac {1}{3} e^{-5 t} \left (c_1 \left (-e^{3 t}\right )+2 c_2 e^{3 t}+c_1+c_2\right ) \\ \end{align*}