81.8.4 problem 11
Internal
problem
ID
[18643]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
VII.
Ordinary
differential
equations
in
two
dependent
variables.
Exercises
at
page
86
Problem
number
:
11
Date
solved
:
Monday, March 31, 2025 at 05:48:42 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d x}y \left (x \right )+3 y \left (x \right )+2 z \left (x \right )&=0\\ \frac {d}{d x}z \left (x \right )+2 y \left (x \right )-4 z \left (x \right )&=0 \end{align*}
✓ Maple. Time used: 0.112 (sec). Leaf size: 85
ode:=[diff(y(x),x)+3*y(x)+2*z(x) = 0, diff(z(x),x)+2*y(x)-4*z(x) = 0];
dsolve(ode);
\begin{align*}
y \left (x \right ) &= c_1 \,{\mathrm e}^{\frac {\left (1+\sqrt {65}\right ) x}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {65}\right ) x}{2}} \\
z \left (x \right ) &= -\frac {c_1 \,{\mathrm e}^{\frac {\left (1+\sqrt {65}\right ) x}{2}} \sqrt {65}}{4}+\frac {c_2 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {65}\right ) x}{2}} \sqrt {65}}{4}-\frac {7 c_1 \,{\mathrm e}^{\frac {\left (1+\sqrt {65}\right ) x}{2}}}{4}-\frac {7 c_2 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {65}\right ) x}{2}}}{4} \\
\end{align*}
✓ Mathematica. Time used: 0.011 (sec). Leaf size: 152
ode={D[y[x],x]+3*y[x]+2*z[x]==0,D[z[x],x]+2*y[x]-4*z[x]==0};
ic={};
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {1}{130} e^{\frac {1}{2} \left (x-\sqrt {65} x\right )} \left (c_1 \left (\left (7 \sqrt {65}-65\right ) e^{\sqrt {65} x}-65-7 \sqrt {65}\right )+4 \sqrt {65} c_2 \left (e^{\sqrt {65} x}-1\right )\right ) \\
z(x)\to \frac {1}{130} e^{\frac {1}{2} \left (x-\sqrt {65} x\right )} \left (c_2 \left (\left (65+7 \sqrt {65}\right ) e^{\sqrt {65} x}+65-7 \sqrt {65}\right )-4 \sqrt {65} c_1 \left (e^{\sqrt {65} x}-1\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.194 (sec). Leaf size: 75
from sympy import *
x = symbols("x")
y = Function("y")
z = Function("z")
ode=[Eq(3*y(x) + 2*z(x) + Derivative(y(x), x),0),Eq(2*y(x) - 4*z(x) + Derivative(z(x), x),0)]
ics = {}
dsolve(ode,func=[y(x),z(x)],ics=ics)
\[
\left [ y{\left (x \right )} = \frac {C_{1} \left (7 - \sqrt {65}\right ) e^{\frac {x \left (1 + \sqrt {65}\right )}{2}}}{4} + \frac {C_{2} \left (7 + \sqrt {65}\right ) e^{\frac {x \left (1 - \sqrt {65}\right )}{2}}}{4}, \ z{\left (x \right )} = C_{1} e^{\frac {x \left (1 + \sqrt {65}\right )}{2}} + C_{2} e^{\frac {x \left (1 - \sqrt {65}\right )}{2}}\right ]
\]