75.32.3 problem 827

Internal problem ID [17280]
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 : 827
Date solved : Tuesday, January 28, 2025 at 09:58:50 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-4 y \left (t \right )+1\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+5 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 33

dsolve([diff(x(t),t)=2*x(t)-4*y(t)+1,diff(y(t),t)=-x(t)+5*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{6 t}+c_{1} {\mathrm e}^{t}-\frac {5}{6} \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{6 t}+\frac {c_{1} {\mathrm e}^{t}}{4}-\frac {1}{6} \\ \end{align*}

✓ Solution by Mathematica

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

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

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