76.9.10 problem 10

Internal problem ID [17520]
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 : 10
Date solved : Tuesday, January 28, 2025 at 10:39:51 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=\frac {5 x \left (t \right )}{4}+\frac {3 y \left (t \right )}{4}\\ \frac {d}{d t}y \left (t \right )&=-\frac {3 x \left (t \right )}{4}-\frac {y \left (t \right )}{4} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 28

dsolve([diff(x(t),t) = 5/4*x(t)+3/4*y(t), diff(y(t),t) = -3/4*x(t)-1/4*y(t), x(0) = 2, y(0) = 3], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\frac {15 t}{4}+2\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{2}} \left (\frac {45 t}{4}-9\right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 40

DSolve[{D[x[t],t]==5/4*x[t]+3/4*y[t],D[y[t],t]==-3/4*x[t]-1/4*y[t]},{x[0]==2,y[0]==3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{t/2} (15 t+8) \\ y(t)\to -\frac {3}{4} e^{t/2} (5 t-4) \\ \end{align*}