76.9.9 problem 9

Internal problem ID [17519]
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 : 9
Date solved : Tuesday, January 28, 2025 at 10:39:50 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 3\\ y \left (0\right ) = -2 \end{align*}

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 28

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\frac {3 t}{2}+3\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{2}} \left (\frac {9 t}{2}+6\right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 38

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

\begin{align*} x(t)\to \frac {3}{2} e^{t/2} (t+2) \\ y(t)\to -\frac {1}{2} e^{t/2} (3 t+4) \\ \end{align*}