71.17.6 problem 13 (a)
Internal
problem
ID
[14491]
Book
:
Ordinary
Differential
Equations
by
Charles
E.
Roberts,
Jr.
CRC
Press.
2010
Section
:
Chapter
7.
Systems
of
First-Order
Differential
Equations.
Exercises
page
329
Problem
number
:
13
(a)
Date
solved
:
Monday, March 31, 2025 at 12:28:32 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d x}y_{1} \left (x \right )&=3 y_{1} \left (x \right )-2 y_{2} \left (x \right )\\ \frac {d}{d x}y_{2} \left (x \right )&=-y_{1} \left (x \right )+y_{2} \left (x \right ) \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) = 1\\ y_{2} \left (0\right ) = -1 \end{align*}
✓ Maple. Time used: 0.164 (sec). Leaf size: 118
ode:=[diff(y__1(x),x) = 3*y__1(x)-2*y__2(x), diff(y__2(x),x) = -y__1(x)+y__2(x)];
ic:=y__1(0) = 1y__2(0) = -1;
dsolve([ode,ic]);
\begin{align*}
y_{1} \left (x \right ) &= \left (\frac {1}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (2+\sqrt {3}\right ) x}+\left (\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-\left (-2+\sqrt {3}\right ) x} \\
y_{2} \left (x \right ) &= -\frac {\left (\frac {1}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (2+\sqrt {3}\right ) x} \sqrt {3}}{2}+\frac {\left (\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-\left (-2+\sqrt {3}\right ) x} \sqrt {3}}{2}+\frac {\left (\frac {1}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (2+\sqrt {3}\right ) x}}{2}+\frac {\left (\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-\left (-2+\sqrt {3}\right ) x}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.009 (sec). Leaf size: 79
ode={D[ y1[x],x]==3*y1[x]-2*y2[x],D[ y2[x],x]==-y1[x]+y2[x]};
ic={y1[0]==1,y2[0]==-1};
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
\begin{align*}
\text {y1}(x)\to \frac {1}{2} e^{-\left (\left (\sqrt {3}-2\right ) x\right )} \left (\left (1+\sqrt {3}\right ) e^{2 \sqrt {3} x}+1-\sqrt {3}\right ) \\
\text {y2}(x)\to -\frac {1}{2} e^{-\left (\left (\sqrt {3}-2\right ) x\right )} \left (e^{2 \sqrt {3} x}+1\right ) \\
\end{align*}
✓ Sympy. Time used: 0.199 (sec). Leaf size: 66
from sympy import *
x = symbols("x")
y__1 = Function("y__1")
y__2 = Function("y__2")
ode=[Eq(-3*y__1(x) + 2*y__2(x) + Derivative(y__1(x), x),0),Eq(y__1(x) - y__2(x) + Derivative(y__2(x), x),0)]
ics = {}
dsolve(ode,func=[y__1(x),y__2(x)],ics=ics)
\[
\left [ y^{1}{\left (x \right )} = - C_{1} \left (1 - \sqrt {3}\right ) e^{x \left (2 - \sqrt {3}\right )} - C_{2} \left (1 + \sqrt {3}\right ) e^{x \left (\sqrt {3} + 2\right )}, \ y^{2}{\left (x \right )} = C_{1} e^{x \left (2 - \sqrt {3}\right )} + C_{2} e^{x \left (\sqrt {3} + 2\right )}\right ]
\]