76.9.6 problem 6

Internal problem ID [17516]
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 : 6
Date solved : Tuesday, January 28, 2025 at 10:39:48 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.079 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 53

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

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