76.9.5 problem 5

Internal problem ID [17515]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.5 (Repeated Eigenvalues). Problems at page 188
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 10:39:47 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )-\frac {y \left (t \right )}{2}\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-3 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.061 (sec). Leaf size: 31

dsolve([diff(x(t),t)=-x(t)-1/2*y(t),diff(y(t),t)=2*x(t)-3*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 49

DSolve[{D[x[t],t]==-x[t]-1/2*y[t],D[y[t],t]==2*x[t]-3*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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