81.8.10 problem 17
Internal
problem
ID
[18649]
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
:
17
Date
solved
:
Monday, March 31, 2025 at 05:48:53 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d x}z \left (x \right )+5 y \left (x \right )-2 z \left (x \right )&=x\\ \frac {d}{d x}y \left (x \right )+4 y \left (x \right )+z \left (x \right )&=x \end{align*}
✓ Maple. Time used: 0.131 (sec). Leaf size: 89
ode:=[diff(z(x),x)+5*y(x)-2*z(x) = x, diff(y(x),x)+4*y(x)+z(x) = x];
dsolve(ode);
\begin{align*}
y \left (x \right ) &= {\mathrm e}^{\left (-1+\sqrt {14}\right ) x} c_2 +{\mathrm e}^{-\left (1+\sqrt {14}\right ) x} c_1 +\frac {3 x}{13}-\frac {7}{169} \\
z \left (x \right ) &= -{\mathrm e}^{\left (-1+\sqrt {14}\right ) x} c_2 \sqrt {14}+{\mathrm e}^{-\left (1+\sqrt {14}\right ) x} c_1 \sqrt {14}-3 \,{\mathrm e}^{\left (-1+\sqrt {14}\right ) x} c_2 -3 \,{\mathrm e}^{-\left (1+\sqrt {14}\right ) x} c_1 +\frac {x}{13}-\frac {11}{169} \\
\end{align*}
✓ Mathematica. Time used: 1.464 (sec). Leaf size: 174
ode={D[z[x],x]+5*y[x]-2*z[x]==x,D[y[x],x]+4*y[x]+z[x]==x};
ic={};
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {3 x}{13}-\frac {1}{28} \left (\left (3 \sqrt {14}-14\right ) c_1+\sqrt {14} c_2\right ) e^{\left (\sqrt {14}-1\right ) x}+\frac {1}{28} \left (\left (14+3 \sqrt {14}\right ) c_1+\sqrt {14} c_2\right ) e^{-\left (\left (1+\sqrt {14}\right ) x\right )}-\frac {7}{169} \\
z(x)\to \frac {x}{13}+\frac {1}{28} \left (5 \sqrt {14} c_1+\left (14-3 \sqrt {14}\right ) c_2\right ) e^{-\left (\left (1+\sqrt {14}\right ) x\right )}+\frac {1}{28} \left (\left (14+3 \sqrt {14}\right ) c_2-5 \sqrt {14} c_1\right ) e^{\left (\sqrt {14}-1\right ) x}-\frac {11}{169} \\
\end{align*}
✓ Sympy. Time used: 0.488 (sec). Leaf size: 83
from sympy import *
x = symbols("x")
y = Function("y")
z = Function("z")
ode=[Eq(-x + 5*y(x) - 2*z(x) + Derivative(z(x), x),0),Eq(-x + 4*y(x) + z(x) + Derivative(y(x), x),0)]
ics = {}
dsolve(ode,func=[y(x),z(x)],ics=ics)
\[
\left [ y{\left (x \right )} = \frac {C_{1} \left (3 - \sqrt {14}\right ) e^{- x \left (1 - \sqrt {14}\right )}}{5} + \frac {C_{2} \left (3 + \sqrt {14}\right ) e^{- x \left (1 + \sqrt {14}\right )}}{5} + \frac {3 x}{13} - \frac {7}{169}, \ z{\left (x \right )} = C_{1} e^{- x \left (1 - \sqrt {14}\right )} + C_{2} e^{- x \left (1 + \sqrt {14}\right )} + \frac {x}{13} - \frac {11}{169}\right ]
\]