87.20.24 problem 25

Internal problem ID [23709]
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 161
Problem number : 25
Date solved : Thursday, October 02, 2025 at 09:44:28 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=y \left (t \right )\\ y^{\prime }\left (t \right )&=-x+2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 24

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= {\mathrm e}^{t} \left (c_2 t +c_1 +c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

ode={D[x[t],t]==y[t],D[y[t],t]==-x[t]+2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^t (c_1 (-t)+c_2 t+c_1)\\ y(t)&\to e^t ((c_2-c_1) t+c_2) \end{align*}

✓ Sympy. Time used: 0.057 (sec). Leaf size: 32

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

\[ \left [ x{\left (t \right )} = - C_{2} t e^{t} - \left (C_{1} - C_{2}\right ) e^{t}, \ y{\left (t \right )} = - C_{1} e^{t} - C_{2} t e^{t}\right ] \]