72.10.5 problem 5

Internal problem ID [14816]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.2. page 277
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 07:17:08 AM
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*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 33

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

\begin{align*} x(t)\to c_1 e^{-t/2} \\ y(t)\to e^{-t/2} (c_1 t+c_2) \\ \end{align*}