81.8.11 problem 18
Internal
problem
ID
[18650]
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
:
18
Date
solved
:
Monday, March 31, 2025 at 05:48:56 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d x}z \left (x \right )+7 y \left (x \right )-9 z \left (x \right )&={\mathrm e}^{x}\\ \frac {d}{d x}y \left (x \right )-y \left (x \right )-3 z \left (x \right )&={\mathrm e}^{2 x} \end{align*}
✓ Maple. Time used: 0.259 (sec). Leaf size: 107
ode:=[diff(z(x),x)+7*y(x)-9*z(x) = exp(x), diff(y(x),x)-y(x)-3*z(x) = exp(2*x)];
dsolve(ode);
\begin{align*}
y \left (x \right ) &= {\mathrm e}^{5 x} \sin \left (\sqrt {5}\, x \right ) c_2 +{\mathrm e}^{5 x} \cos \left (\sqrt {5}\, x \right ) c_1 +\frac {{\mathrm e}^{x}}{7}-\frac {{\mathrm e}^{2 x}}{2} \\
z \left (x \right ) &= -\frac {{\mathrm e}^{2 x}}{2}+\frac {4 \,{\mathrm e}^{5 x} \sin \left (\sqrt {5}\, x \right ) c_2}{3}+\frac {{\mathrm e}^{5 x} \sqrt {5}\, \cos \left (\sqrt {5}\, x \right ) c_2}{3}+\frac {4 \,{\mathrm e}^{5 x} \cos \left (\sqrt {5}\, x \right ) c_1}{3}-\frac {{\mathrm e}^{5 x} \sqrt {5}\, \sin \left (\sqrt {5}\, x \right ) c_1}{3} \\
\end{align*}
✓ Mathematica. Time used: 0.653 (sec). Leaf size: 123
ode={D[z[x],x]+7*y[x]-9*z[x]==Exp[x],D[y[x],x]-y[x]-3*z[x]==Exp[2*x]};
ic={};
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{14} e^x \left (2-7 e^x\right )+c_1 e^{5 x} \cos \left (\sqrt {5} x\right )-\frac {(4 c_1-3 c_2) e^{5 x} \sin \left (\sqrt {5} x\right )}{\sqrt {5}} \\
z(x)\to -\frac {e^{2 x}}{2}+c_2 e^{5 x} \cos \left (\sqrt {5} x\right )-\frac {(7 c_1-4 c_2) e^{5 x} \sin \left (\sqrt {5} x\right )}{\sqrt {5}} \\
\end{align*}
✓ Sympy. Time used: 0.895 (sec). Leaf size: 187
from sympy import *
x = symbols("x")
y = Function("y")
z = Function("z")
ode=[Eq(7*y(x) - 9*z(x) - exp(x) + Derivative(z(x), x),0),Eq(-y(x) - 3*z(x) - exp(2*x) + Derivative(y(x), x),0)]
ics = {}
dsolve(ode,func=[y(x),z(x)],ics=ics)
\[
\left [ y{\left (x \right )} = - \left (\frac {4 C_{1}}{7} - \frac {\sqrt {5} C_{2}}{7}\right ) e^{5 x} \sin {\left (\sqrt {5} x \right )} + \left (\frac {\sqrt {5} C_{1}}{7} + \frac {4 C_{2}}{7}\right ) e^{5 x} \cos {\left (\sqrt {5} x \right )} - \frac {e^{2 x} \sin ^{2}{\left (\sqrt {5} x \right )}}{2} - \frac {e^{2 x} \cos ^{2}{\left (\sqrt {5} x \right )}}{2} + \frac {e^{x} \sin ^{2}{\left (\sqrt {5} x \right )}}{7} + \frac {e^{x} \cos ^{2}{\left (\sqrt {5} x \right )}}{7}, \ z{\left (x \right )} = - C_{1} e^{5 x} \sin {\left (\sqrt {5} x \right )} + C_{2} e^{5 x} \cos {\left (\sqrt {5} x \right )} - \frac {e^{2 x} \sin ^{2}{\left (\sqrt {5} x \right )}}{2} - \frac {e^{2 x} \cos ^{2}{\left (\sqrt {5} x \right )}}{2}\right ]
\]