76.9.1 problem 1

Internal problem ID [17511]
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 3.5 (Repeated Eigenvalues). Problems at page 188
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 10:39:44 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 41

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

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