87.21.31 problem 32
Internal
problem
ID
[23745]
Book
:
Ordinary
differential
equations
with
modern
applications.
Ladas,
G.
E.
and
Finizio,
N.
Wadsworth
Publishing.
California.
1978.
ISBN
0-534-00552-7.
QA372.F56
Section
:
Chapter
3.
Linear
Systems.
Exercise
at
page
178
Problem
number
:
32
Date
solved
:
Thursday, October 02, 2025 at 09:44:50 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }&=-2 x+y \left (t \right )\\ y^{\prime }\left (t \right )&=x-2 y \left (t \right )\\ z^{\prime }\left (t \right )&=x+y \left (t \right )-5 z \left (t \right )\\ u^{\prime }\left (t \right )&=5 z \left (t \right ) \end{align*}
With initial conditions
\begin{align*}
x \left (0\right )&=1 \\
y \left (0\right )&=0 \\
z \left (0\right )&=0 \\
u \left (0\right )&=0 \\
\end{align*}
✓ Maple. Time used: 0.106 (sec). Leaf size: 66
ode:=[diff(x(t),t) = -2*x(t)+y(t), diff(y(t),t) = x(t)-2*y(t), diff(z(t),t) = x(t)+y(t)-5*z(t), diff(u(t),t) = 5*z(t)];
ic:=[x(0) = 1, y(0) = 0, z(0) = 0, u(0) = 0];
dsolve([ode,op(ic)]);
\begin{align*}
u \left (t \right ) &= \frac {{\mathrm e}^{-5 t}}{4}-\frac {5 \,{\mathrm e}^{-t}}{4}+1 \\
x \left (t \right ) &= \frac {{\mathrm e}^{-3 t}}{2}+\frac {{\mathrm e}^{-t}}{2} \\
y \left (t \right ) &= -\frac {{\mathrm e}^{-3 t}}{2}+\frac {{\mathrm e}^{-t}}{2} \\
z \left (t \right ) &= \frac {{\mathrm e}^{-t}}{4}-\frac {{\mathrm e}^{-5 t}}{4} \\
\end{align*}
✓ Mathematica. Time used: 0.017 (sec). Leaf size: 227
ode={D[x[t],t]==2*x[t]+y[t],D[y[t],t]==x[t]-2*y[t],D[z[t],t]==x[t]+y[t]-5*z[t],D[u[t],t]==5*z[t]};
ic={x[0]==1,y[0]==0,z[0]==0,u[0]==0};
DSolve[{ode,ic},{x[t],y[t],z[t],u[t]},t,IncludeSingularSolutions->True]
\begin{align*} u(t)&\to \frac {1}{20} e^{-\left (\left (5+\sqrt {5}\right ) t\right )} \left (-\left (\left (\sqrt {5}-5\right ) e^{5 t}\right )+2 e^{\sqrt {5} t}-12 e^{\left (5+\sqrt {5}\right ) t}+\left (5+\sqrt {5}\right ) e^{\left (5+2 \sqrt {5}\right ) t}\right )\\ x(t)&\to \frac {1}{10} e^{-\sqrt {5} t} \left (\left (5+2 \sqrt {5}\right ) e^{2 \sqrt {5} t}+5-2 \sqrt {5}\right )\\ y(t)&\to \frac {e^{-\sqrt {5} t} \left (e^{2 \sqrt {5} t}-1\right )}{2 \sqrt {5}}\\ z(t)&\to -\frac {1}{20} e^{-\left (\left (5+\sqrt {5}\right ) t\right )} \left (\left (\sqrt {5}-1\right ) e^{5 t}+2 e^{\sqrt {5} t}-\left (1+\sqrt {5}\right ) e^{\left (5+2 \sqrt {5}\right ) t}\right ) \end{align*}
✓ Sympy. Time used: 0.290 (sec). Leaf size: 163
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
u = Function("u")
ode=[Eq(-2*x(t) - y(t) + Derivative(x(t), t),0),Eq(-x(t) + 2*y(t) + Derivative(y(t), t),0),Eq(-x(t) - y(t) + 5*z(t) + Derivative(z(t), t),0),Eq(-5*z(t) + Derivative(u(t), t),0)]
ics = {x(0): 1, y(0): 0, z(0): 0, u(0): 0}
dsolve(ode,func=[x(t),y(t),z(t),u(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {\left (2 \sqrt {5} + 5\right ) e^{\sqrt {5} t}}{10} + \frac {\left (5 - 2 \sqrt {5}\right ) e^{- \sqrt {5} t}}{10}, \ y{\left (t \right )} = \frac {\sqrt {5} e^{\sqrt {5} t}}{10} - \frac {\sqrt {5} e^{- \sqrt {5} t}}{10}, \ z{\left (t \right )} = \frac {\left (1 + \sqrt {5}\right ) e^{\sqrt {5} t}}{20} + \frac {\left (1 - \sqrt {5}\right ) e^{- \sqrt {5} t}}{20} - \frac {e^{- 5 t}}{10}, \ u{\left (t \right )} = \frac {\left (\sqrt {5} + 5\right ) e^{\sqrt {5} t}}{20} - \frac {3}{5} + \frac {\left (5 - \sqrt {5}\right ) e^{- \sqrt {5} t}}{20} + \frac {e^{- 5 t}}{10}\right ]
\]