81.8.5 problem 12
Internal
problem
ID
[18644]
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
:
12
Date
solved
:
Monday, March 31, 2025 at 05:48:44 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 )+y \left (x \right )-2 z \left (x \right )&=0 \end{align*}
✓ Maple. Time used: 0.131 (sec). Leaf size: 81
ode:=[diff(y(x),x)-3*y(x)-2*z(x) = 0, diff(z(x),x)+y(x)-2*z(x) = 0];
dsolve(ode);
\begin{align*}
y \left (x \right ) &= {\mathrm e}^{\frac {5 x}{2}} \left (\sin \left (\frac {\sqrt {7}\, x}{2}\right ) c_1 +\cos \left (\frac {\sqrt {7}\, x}{2}\right ) c_2 \right ) \\
z \left (x \right ) &= -\frac {{\mathrm e}^{\frac {5 x}{2}} \left (\sin \left (\frac {\sqrt {7}\, x}{2}\right ) \sqrt {7}\, c_2 -\cos \left (\frac {\sqrt {7}\, x}{2}\right ) \sqrt {7}\, c_1 +\sin \left (\frac {\sqrt {7}\, x}{2}\right ) c_1 +\cos \left (\frac {\sqrt {7}\, x}{2}\right ) c_2 \right )}{4} \\
\end{align*}
✓ Mathematica. Time used: 0.016 (sec). Leaf size: 111
ode={D[y[x],x]-3*y[x]-2*z[x]==0,D[z[x],x]+y[x]-2*z[x]==0};
ic={};
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{7} e^{5 x/2} \left (7 c_1 \cos \left (\frac {\sqrt {7} x}{2}\right )+\sqrt {7} (c_1+4 c_2) \sin \left (\frac {\sqrt {7} x}{2}\right )\right ) \\
z(x)\to \frac {1}{7} e^{5 x/2} \left (7 c_2 \cos \left (\frac {\sqrt {7} x}{2}\right )-\sqrt {7} (2 c_1+c_2) \sin \left (\frac {\sqrt {7} x}{2}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.182 (sec). Leaf size: 99
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(y(x) - 2*z(x) + Derivative(z(x), x),0)]
ics = {}
dsolve(ode,func=[y(x),z(x)],ics=ics)
\[
\left [ y{\left (x \right )} = - \left (\frac {C_{1}}{2} - \frac {\sqrt {7} C_{2}}{2}\right ) e^{\frac {5 x}{2}} \cos {\left (\frac {\sqrt {7} x}{2} \right )} + \left (\frac {\sqrt {7} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{\frac {5 x}{2}} \sin {\left (\frac {\sqrt {7} x}{2} \right )}, \ z{\left (x \right )} = C_{1} e^{\frac {5 x}{2}} \cos {\left (\frac {\sqrt {7} x}{2} \right )} - C_{2} e^{\frac {5 x}{2}} \sin {\left (\frac {\sqrt {7} x}{2} \right )}\right ]
\]