81.8.12 problem 19
Internal
problem
ID
[18651]
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
:
19
Date
solved
:
Monday, March 31, 2025 at 05:48:58 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d x}y \left (x \right )-2 y \left (x \right )-2 z \left (x \right )&={\mathrm e}^{3 x}\\ \frac {d}{d x}z \left (x \right )+5 y \left (x \right )-2 z \left (x \right )&={\mathrm e}^{4 x} \end{align*}
✓ Maple. Time used: 0.270 (sec). Leaf size: 89
ode:=[diff(y(x),x)-2*y(x)-2*z(x) = exp(3*x), diff(z(x),x)+5*y(x)-2*z(x) = exp(4*x)];
dsolve(ode);
\begin{align*}
y \left (x \right ) &= {\mathrm e}^{2 x} \sin \left (\sqrt {10}\, x \right ) c_2 +{\mathrm e}^{2 x} \cos \left (\sqrt {10}\, x \right ) c_1 +\frac {{\mathrm e}^{4 x}}{7}+\frac {{\mathrm e}^{3 x}}{11} \\
z \left (x \right ) &= -\frac {5 \,{\mathrm e}^{3 x}}{11}+\frac {{\mathrm e}^{2 x} \sqrt {10}\, \cos \left (\sqrt {10}\, x \right ) c_2}{2}-\frac {{\mathrm e}^{2 x} \sqrt {10}\, \sin \left (\sqrt {10}\, x \right ) c_1}{2}+\frac {{\mathrm e}^{4 x}}{7} \\
\end{align*}
✓ Mathematica. Time used: 0.248 (sec). Leaf size: 110
ode={D[y[x],x]-2*y[x]-2*z[x]==Exp[3*x],D[z[x],x]+5*y[x]-2*z[x]==Exp[4*x]};
ic={};
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{385} e^{2 x} \left (5 e^x \left (11 e^x+7\right )+385 c_1 \cos \left (\sqrt {10} x\right )+77 \sqrt {10} c_2 \sin \left (\sqrt {10} x\right )\right ) \\
z(x)\to \frac {1}{154} e^{2 x} \left (2 e^x \left (11 e^x-35\right )+154 c_2 \cos \left (\sqrt {10} x\right )-77 \sqrt {10} c_1 \sin \left (\sqrt {10} x\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.565 (sec). Leaf size: 218
from sympy import *
x = symbols("x")
y = Function("y")
z = Function("z")
ode=[Eq(-2*y(x) - 2*z(x) - exp(3*x) + Derivative(y(x), x),0),Eq(5*y(x) - 2*z(x) - exp(4*x) + Derivative(z(x), x),0)]
ics = {}
dsolve(ode,func=[y(x),z(x)],ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\sqrt {10} C_{1} e^{2 x} \sin {\left (\sqrt {10} x \right )}}{5} + \frac {\sqrt {10} C_{2} e^{2 x} \cos {\left (\sqrt {10} x \right )}}{5} + \frac {e^{4 x} \sin ^{2}{\left (\sqrt {10} x \right )}}{7} + \frac {e^{4 x} \cos ^{2}{\left (\sqrt {10} x \right )}}{7} + \frac {e^{3 x} \sin ^{2}{\left (\sqrt {10} x \right )}}{11} + \frac {e^{3 x} \cos ^{2}{\left (\sqrt {10} x \right )}}{11}, \ z{\left (x \right )} = C_{1} e^{2 x} \cos {\left (\sqrt {10} x \right )} - C_{2} e^{2 x} \sin {\left (\sqrt {10} x \right )} + \frac {e^{4 x} \sin ^{2}{\left (\sqrt {10} x \right )}}{7} + \frac {e^{4 x} \cos ^{2}{\left (\sqrt {10} x \right )}}{7} - \frac {5 e^{3 x} \sin ^{2}{\left (\sqrt {10} x \right )}}{11} - \frac {5 e^{3 x} \cos ^{2}{\left (\sqrt {10} x \right )}}{11}\right ]
\]