52.10.14 problem 13

Internal problem ID [8408]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 13
Date solved : Monday, January 27, 2025 at 03:57:40 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 4\\ y \left (0\right ) = 5 \end{align*}

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 24

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 32

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

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