76.9.12 problem 12

Internal problem ID [17522]
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 : 12
Date solved : Tuesday, January 28, 2025 at 10:39:53 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*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 3 \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 28

dsolve([diff(x(t),t) = 2*x(t)+1/2*y(t), diff(y(t),t) = -1/2*x(t)+y(t), x(0) = 1, y(0) = 3], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 34

DSolve[{D[x[t],t]==2*x[t]+1/2*y[t],D[y[t],t]==-1/2*x[t]+y[t]},{x[0]==1,y[0]==3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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