71.18.3 problem 3
Internal
problem
ID
[14506]
Book
:
Ordinary
Differential
Equations
by
Charles
E.
Roberts,
Jr.
CRC
Press.
2010
Section
:
Chapter
8.
Linear
Systems
of
First-Order
Differential
Equations.
Exercises
8.3
page
379
Problem
number
:
3
Date
solved
:
Monday, March 31, 2025 at 12:28:59 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d x}y_{1} \left (x \right )&=2 y_{2} \left (x \right )\\ \frac {d}{d x}y_{2} \left (x \right )&=3 y_{1} \left (x \right )\\ \frac {d}{d x}y_{3} \left (x \right )&=2 y_{3} \left (x \right )-y_{1} \left (x \right ) \end{align*}
✓ Maple. Time used: 0.161 (sec). Leaf size: 106
ode:=[diff(y__1(x),x) = 2*y__2(x), diff(y__2(x),x) = 3*y__1(x), diff(y__3(x),x) = 2*y__3(x)-y__1(x)];
dsolve(ode);
\begin{align*}
y_{1} \left (x \right ) &= c_2 \,{\mathrm e}^{\sqrt {6}\, x}+c_3 \,{\mathrm e}^{-\sqrt {6}\, x} \\
y_{2} \left (x \right ) &= \frac {\sqrt {6}\, \left (c_2 \,{\mathrm e}^{\sqrt {6}\, x}-c_3 \,{\mathrm e}^{-\sqrt {6}\, x}\right )}{2} \\
y_{3} \left (x \right ) &= \frac {2 \,{\mathrm e}^{2 x} c_1}{\left (-2+\sqrt {6}\right ) \left (2+\sqrt {6}\right )}+\frac {{\mathrm e}^{-\sqrt {6}\, x} c_3}{2+\sqrt {6}}-\frac {{\mathrm e}^{\sqrt {6}\, x} c_2}{-2+\sqrt {6}} \\
\end{align*}
✓ Mathematica. Time used: 0.02 (sec). Leaf size: 232
ode={D[ y1[x],x]==2*y2[x],D[ y2[x],x]==3*y1[x],D[ y3[x],x]==2*y3[x]-y1[x]};
ic={};
DSolve[{ode,ic},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions->True]
\begin{align*}
\text {y1}(x)\to \frac {1}{6} e^{-\sqrt {6} x} \left (3 c_1 \left (e^{2 \sqrt {6} x}+1\right )+\sqrt {6} c_2 \left (e^{2 \sqrt {6} x}-1\right )\right ) \\
\text {y2}(x)\to \frac {1}{4} e^{-\sqrt {6} x} \left (\sqrt {6} c_1 \left (e^{2 \sqrt {6} x}-1\right )+2 c_2 \left (e^{2 \sqrt {6} x}+1\right )\right ) \\
\text {y3}(x)\to \frac {1}{12} e^{-\sqrt {6} x} \left (2 \left (c_2 \left (-\left (3+\sqrt {6}\right ) e^{2 \sqrt {6} x}+6 e^{\left (2+\sqrt {6}\right ) x}-3+\sqrt {6}\right )+6 c_3 e^{\left (2+\sqrt {6}\right ) x}\right )-3 c_1 \left (\left (2+\sqrt {6}\right ) e^{2 \sqrt {6} x}-4 e^{\left (2+\sqrt {6}\right ) x}+2-\sqrt {6}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.245 (sec). Leaf size: 102
from sympy import *
x = symbols("x")
y__1 = Function("y__1")
y__2 = Function("y__2")
y__3 = Function("y__3")
ode=[Eq(-2*y__2(x) + Derivative(y__1(x), x),0),Eq(-3*y__1(x) + Derivative(y__2(x), x),0),Eq(y__1(x) - 2*y__3(x) + Derivative(y__3(x), x),0)]
ics = {}
dsolve(ode,func=[y__1(x),y__2(x),y__3(x)],ics=ics)
\[
\left [ y^{1}{\left (x \right )} = C_{1} \left (2 - \sqrt {6}\right ) e^{\sqrt {6} x} + C_{2} \left (2 + \sqrt {6}\right ) e^{- \sqrt {6} x}, \ y^{2}{\left (x \right )} = - C_{1} \left (3 - \sqrt {6}\right ) e^{\sqrt {6} x} - C_{2} \left (\sqrt {6} + 3\right ) e^{- \sqrt {6} x}, \ y^{3}{\left (x \right )} = C_{1} e^{\sqrt {6} x} + C_{2} e^{- \sqrt {6} x} + C_{3} e^{2 x}\right ]
\]