76.6.15 problem 17

Internal problem ID [17470]
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 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 10:39:11 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-\frac {x \left (t \right )}{4}-\frac {3 y \left (t \right )}{4}+8\\ \frac {d}{d t}y \left (t \right )&=\frac {x \left (t \right )}{2}+y \left (t \right )-\frac {23}{2} \end{align*}

✓ Solution by Maple

Time used: 0.111 (sec). Leaf size: 38

dsolve([diff(x(t),t)=-1/4*x(t)-75/100*y(t)+8,diff(y(t),t)=1/2*x(t)+y(t)-115/10],singsol=all)
 

\begin{align*} x \left (t \right ) &= 4 \,{\mathrm e}^{\frac {t}{2}} c_{1} +{\mathrm e}^{\frac {t}{4}} c_{2} +5 \\ y \left (t \right ) &= -4 \,{\mathrm e}^{\frac {t}{2}} c_{1} -\frac {2 \,{\mathrm e}^{\frac {t}{4}} c_{2}}{3}+9 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 75

DSolve[{D[x[t],t]==-1/4*x[t]-75/100*y[t]+8,D[y[t],t]==1/2*x[t]+y[t]-115/10},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to 3 (c_1+c_2) e^{t/4}-(2 c_1+3 c_2) e^{t/2}+5 \\ y(t)\to -2 (c_1+c_2) e^{t/4}+(2 c_1+3 c_2) e^{t/2}+9 \\ \end{align*}