69.1.132 problem 191

Internal problem ID [14285]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 191
Date solved : Tuesday, January 28, 2025 at 06:25:44 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right )+6 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 77

dsolve([diff(x(t),t)=2*x(t)-3*y(t),diff(y(t),t)=5*x(t)+6*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} \left (\sin \left (\sqrt {11}\, t \right ) c_{1} +\cos \left (\sqrt {11}\, t \right ) c_{2} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{4 t} \left (\sqrt {11}\, \sin \left (\sqrt {11}\, t \right ) c_{2} -\sqrt {11}\, \cos \left (\sqrt {11}\, t \right ) c_{1} -2 \sin \left (\sqrt {11}\, t \right ) c_{1} -2 \cos \left (\sqrt {11}\, t \right ) c_{2} \right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 99

DSolve[{D[x[t],t]==2*x[t]-3*y[t],D[y[t],t]==5*x[t]+6*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_1 e^{4 t} \cos \left (\sqrt {11} t\right )-\frac {(2 c_1+3 c_2) e^{4 t} \sin \left (\sqrt {11} t\right )}{\sqrt {11}} \\ y(t)\to c_2 e^{4 t} \cos \left (\sqrt {11} t\right )+\frac {(5 c_1+2 c_2) e^{4 t} \sin \left (\sqrt {11} t\right )}{\sqrt {11}} \\ \end{align*}