76.10.5 problem 5

Internal problem ID [17527]
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 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 10:39:57 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=8 x \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: 0.008 (sec). Leaf size: 33

dsolve([diff(x(t),t) = 2*y(t), diff(y(t),t) = 8*x(t), x(0) = 1, y(0) = -3], singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {5 \,{\mathrm e}^{-4 t}}{4}-\frac {{\mathrm e}^{4 t}}{4} \\ y \left (t \right ) &= -\frac {5 \,{\mathrm e}^{-4 t}}{2}-\frac {{\mathrm e}^{4 t}}{2} \\ \end{align*}

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]==2*y[t],D[y[t],t]==8*x[t]},{x[0]==1,y[0]==-3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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