76.9.3 problem 3

Internal problem ID [17513]
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 : 3
Date solved : Tuesday, January 28, 2025 at 10:39:46 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.071 (sec). Leaf size: 31

dsolve([diff(x(t),t)=-3/2*x(t)+y(t),diff(y(t),t)=-1/4*x(t)-1/2*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 54

DSolve[{D[x[t],t]==-3/2*x[t]+y[t],D[y[t],t]==-1/4*x[t]-1/2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-t} (2 c_2 t-c_1 (t-2)) \\ y(t)\to \frac {1}{4} e^{-t} (c_1 (-t)+2 c_2 t+4 c_2) \\ \end{align*}