75.29.5 problem 806

Internal problem ID [17259]
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 22. Integration of homogeneous linear systems with constant coefficients. Eulers method. Exercises page 230
Problem number : 806
Date solved : Tuesday, January 28, 2025 at 09:58:32 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 27

dsolve([diff(x(t),t) = 4*x(t)-5*y(t), diff(y(t),t) = x(t), x(0) = 0, y(0) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= -5 \,{\mathrm e}^{2 t} \sin \left (t \right ) \\ y \left (t \right ) &= {\mathrm e}^{2 t} \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 27

DSolve[{D[x[t],t]==4*x[t]-4*y[t],D[y[t],t]==x[t]},{x[0]==0,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -4 e^{2 t} t \\ y(t)\to e^{2 t} (1-2 t) \\ \end{align*}