23.4.6 problem 8(f)

Internal problem ID [4171]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 8(f)
Date solved : Tuesday, March 04, 2025 at 05:54:51 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y_{1} \left (x \right )&=y_{2} \left (x \right )\\ \frac {d}{d x}y_{2} \left (x \right )&=y_{1} \left (x \right ) \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 30

ode:=[diff(y__1(x),x) = y__2(x), diff(y__2(x),x) = y__1(x)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{x} \\ y_{2} \left (x \right ) &= -{\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{x} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 68

ode={D[y1[x],x]==y2[x],D[y2[x],x]==y1[x]}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)\to \frac {1}{2} e^{-x} \left (c_1 \left (e^{2 x}+1\right )+c_2 \left (e^{2 x}-1\right )\right ) \\ \text {y2}(x)\to \frac {1}{2} e^{-x} \left (c_1 \left (e^{2 x}-1\right )+c_2 \left (e^{2 x}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.079 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-y__2(x) + Derivative(y__1(x), x),0),Eq(-y__1(x) + Derivative(y__2(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x)],ics=ics)
 

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