52.10.4 problem 4

Internal problem ID [8398]
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 : 4
Date solved : Monday, January 27, 2025 at 03:57:31 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-\frac {5 x \left (t \right )}{2}+2 y\\ y^{\prime }&=\frac {3 x \left (t \right )}{4}-2 y \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 35

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{-\frac {7 t}{2}} \\ y &= \frac {3 \,{\mathrm e}^{-t} c_{1}}{4}-\frac {c_{2} {\mathrm e}^{-\frac {7 t}{2}}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 165

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

\begin{align*} x(t)\to \frac {1}{210} e^{\frac {1}{4} \left (t-\sqrt {105} t\right )} \left (3 c_1 \left (\left (35+3 \sqrt {105}\right ) e^{\frac {\sqrt {105} t}{2}}+35-3 \sqrt {105}\right )+8 \sqrt {105} c_2 \left (e^{\frac {\sqrt {105} t}{2}}-1\right )\right ) \\ y(t)\to \frac {1}{70} e^{\frac {1}{4} \left (t-\sqrt {105} t\right )} \left (\sqrt {105} c_1 \left (e^{\frac {\sqrt {105} t}{2}}-1\right )-c_2 \left (\left (3 \sqrt {105}-35\right ) e^{\frac {\sqrt {105} t}{2}}-35-3 \sqrt {105}\right )\right ) \\ \end{align*}