72.14.14 problem 18
Internal
problem
ID
[14877]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
3.
Linear
Systems.
Exercises
section
3.8
page
371
Problem
number
:
18
Date
solved
:
Tuesday, January 28, 2025 at 07:18:02 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=-10 x \left (t \right )+10 y\\ y^{\prime }&=28 x \left (t \right )-y\\ z^{\prime }\left (t \right )&=-\frac {8 z \left (t \right )}{3} \end{align*}
✓ Solution by Maple
Time used: 0.076 (sec). Leaf size: 94
dsolve([diff(x(t),t)=-10*x(t)+10*y(t)+0*z(t),diff(y(t),t)=28*x(t)-1*y(t)+0*z(t),diff(z(t),t)=0*x(t)+0*y(t)-8/3*z(t)],singsol=all)
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-11+\sqrt {1201}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (11+\sqrt {1201}\right ) t}{2}} \\
y &= \frac {c_{1} {\mathrm e}^{\frac {\left (-11+\sqrt {1201}\right ) t}{2}} \sqrt {1201}}{20}-\frac {c_{2} {\mathrm e}^{-\frac {\left (11+\sqrt {1201}\right ) t}{2}} \sqrt {1201}}{20}+\frac {9 c_{1} {\mathrm e}^{\frac {\left (-11+\sqrt {1201}\right ) t}{2}}}{20}+\frac {9 c_{2} {\mathrm e}^{-\frac {\left (11+\sqrt {1201}\right ) t}{2}}}{20} \\
z &= c_{3} {\mathrm e}^{-\frac {8 t}{3}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.028 (sec). Leaf size: 312
DSolve[{D[x[t],t]==-10*x[t]+10*y[t]+0*z[t],D[y[t],t]==28*x[t]-1*y[t]+0*z[t],D[z[t],t]==0*x[t]+0*y[t]-8/3*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to \frac {e^{-\frac {1}{2} \left (11+\sqrt {1201}\right ) t} \left (c_1 \left (\left (1201-9 \sqrt {1201}\right ) e^{\sqrt {1201} t}+1201+9 \sqrt {1201}\right )+20 \sqrt {1201} c_2 \left (e^{\sqrt {1201} t}-1\right )\right )}{2402} \\
y(t)\to \frac {e^{-\frac {1}{2} \left (11+\sqrt {1201}\right ) t} \left (56 \sqrt {1201} c_1 \left (e^{\sqrt {1201} t}-1\right )+c_2 \left (\left (1201+9 \sqrt {1201}\right ) e^{\sqrt {1201} t}+1201-9 \sqrt {1201}\right )\right )}{2402} \\
z(t)\to c_3 e^{-8 t/3} \\
x(t)\to \frac {e^{-\frac {1}{2} \left (11+\sqrt {1201}\right ) t} \left (c_1 \left (\left (1201-9 \sqrt {1201}\right ) e^{\sqrt {1201} t}+1201+9 \sqrt {1201}\right )+20 \sqrt {1201} c_2 \left (e^{\sqrt {1201} t}-1\right )\right )}{2402} \\
y(t)\to \frac {e^{-\frac {1}{2} \left (11+\sqrt {1201}\right ) t} \left (56 \sqrt {1201} c_1 \left (e^{\sqrt {1201} t}-1\right )+c_2 \left (\left (1201+9 \sqrt {1201}\right ) e^{\sqrt {1201} t}+1201-9 \sqrt {1201}\right )\right )}{2402} \\
z(t)\to 0 \\
\end{align*}