75.32.1 problem 825

Internal problem ID [17278]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 23.3 dAlemberts method. Exercises page 243
Problem number : 825
Date solved : Tuesday, January 28, 2025 at 09:58:48 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 31

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

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{6 t}+c_{2} {\mathrm e}^{t} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{6 t}}{4}-c_{2} {\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 67

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

\begin{align*} x(t)\to \frac {1}{5} e^t \left (c_1 \left (4 e^{5 t}+1\right )+4 c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} e^t \left (c_1 \left (e^{5 t}-1\right )+c_2 \left (e^{5 t}+4\right )\right ) \\ \end{align*}