63.1.132 problem 191

Internal problem ID [15572]
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 : Thursday, October 02, 2025 at 10:20:16 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*}

✓ Maple. Time used: 0.137 (sec). Leaf size: 77

ode:=[diff(x(t),t) = 2*x(t)-3*y(t), diff(y(t),t) = 5*x(t)+6*y(t)]; 
dsolve(ode);
 

\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 (\sin \left (\sqrt {11}\, t \right ) \sqrt {11}\, c_2 -\cos \left (\sqrt {11}\, t \right ) \sqrt {11}\, c_1 -2 \sin \left (\sqrt {11}\, t \right ) c_1 -2 \cos \left (\sqrt {11}\, t \right ) c_2 \right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 99

ode={D[x[t],t]==2*x[t]-3*y[t],D[y[t],t]==5*x[t]+6*y[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.121 (sec). Leaf size: 90

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + 3*y(t) + Derivative(x(t), t),0),Eq(-5*x(t) - 6*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {2 C_{1}}{5} + \frac {\sqrt {11} C_{2}}{5}\right ) e^{4 t} \cos {\left (\sqrt {11} t \right )} - \left (\frac {\sqrt {11} C_{1}}{5} - \frac {2 C_{2}}{5}\right ) e^{4 t} \sin {\left (\sqrt {11} t \right )}, \ y{\left (t \right )} = C_{1} e^{4 t} \cos {\left (\sqrt {11} t \right )} - C_{2} e^{4 t} \sin {\left (\sqrt {11} t \right )}\right ] \]