87.21.10 problem 10

Internal problem ID [23724]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 178
Problem number : 10
Date solved : Thursday, October 02, 2025 at 09:44:37 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=3 x-5 y \left (t \right )\\ y^{\prime }\left (t \right )&=4 x-5 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (\pi \right )&=1 \\ y \left (\pi \right )&={\frac {4}{5}} \\ \end{align*}

✓ Maple. Time used: 0.087 (sec). Leaf size: 63

ode:=[diff(x(t),t) = 3*x(t)-5*y(t), diff(y(t),t) = 4*x(t)-5*y(t)]; 
ic:=[x(Pi) = 1, y(Pi) = 4/5]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 42

ode={D[x[t],t]==3*x[t]-5*y[t],D[y[t],t]==4*x[t]-5*y[t] }; 
ic={x[Pi]==1,y[Pi]==4/5}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{\pi -t} \cos (2 t)\\ y(t)&\to \frac {2}{5} e^{\pi -t} (\sin (2 t)+2 \cos (2 t)) \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(-4*x(t) + 5*y(t) + Derivative(y(t), t),0)] 
ics = {x(pi): 1, y(pi): 4/5} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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