61.5.13 problem Problem 3(b)
Internal
problem
ID
[15380]
Book
:
APPLIED
DIFFERENTIAL
EQUATIONS
The
Primary
Course
by
Vladimir
A.
Dobrushkin.
CRC
Press
2015
Section
:
Chapter
6.
Introduction
to
Systems
of
ODEs.
Problems
page
408
Problem
number
:
Problem
3(b)
Date
solved
:
Thursday, October 02, 2025 at 10:12:23 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=\frac {5 x \left (t \right )}{4}+\frac {3 y}{4}\\ y^{\prime }&=\frac {x \left (t \right )}{2}-\frac {3 y}{2} \end{align*}
✓ Maple. Time used: 0.115 (sec). Leaf size: 85
ode:=[diff(x(t),t) = 5/4*x(t)+3/4*y(t), diff(y(t),t) = 1/2*x(t)-3/2*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (-1+\sqrt {145}\right ) t}{8}}+c_2 \,{\mathrm e}^{-\frac {\left (1+\sqrt {145}\right ) t}{8}} \\
y \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{\frac {\left (-1+\sqrt {145}\right ) t}{8}} \sqrt {145}}{6}-\frac {c_2 \,{\mathrm e}^{-\frac {\left (1+\sqrt {145}\right ) t}{8}} \sqrt {145}}{6}-\frac {11 c_1 \,{\mathrm e}^{\frac {\left (-1+\sqrt {145}\right ) t}{8}}}{6}-\frac {11 c_2 \,{\mathrm e}^{-\frac {\left (1+\sqrt {145}\right ) t}{8}}}{6} \\
\end{align*}
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 161
ode={D[x[t],t]==5/4*x[t]+3/4*y[t],D[y[t],t]==1/2*x[t]-3/2*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{290} e^{-\frac {1}{8} \left (1+\sqrt {145}\right ) t} \left (c_1 \left (\left (145+11 \sqrt {145}\right ) e^{\frac {\sqrt {145} t}{4}}+145-11 \sqrt {145}\right )+6 \sqrt {145} c_2 \left (e^{\frac {\sqrt {145} t}{4}}-1\right )\right )\\ y(t)&\to \frac {1}{290} e^{-\frac {1}{8} \left (1+\sqrt {145}\right ) t} \left (4 \sqrt {145} c_1 \left (e^{\frac {\sqrt {145} t}{4}}-1\right )-c_2 \left (\left (11 \sqrt {145}-145\right ) e^{\frac {\sqrt {145} t}{4}}-145-11 \sqrt {145}\right )\right ) \end{align*}
✓ Sympy. Time used: 0.125 (sec). Leaf size: 75
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-5*x(t)/4 - 3*y(t)/4 + Derivative(x(t), t),0),Eq(-x(t)/2 + 3*y(t)/2 + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {C_{1} \left (11 - \sqrt {145}\right ) e^{- \frac {t \left (1 + \sqrt {145}\right )}{8}}}{4} + \frac {C_{2} \left (11 + \sqrt {145}\right ) e^{- \frac {t \left (1 - \sqrt {145}\right )}{8}}}{4}, \ y{\left (t \right )} = C_{1} e^{- \frac {t \left (1 + \sqrt {145}\right )}{8}} + C_{2} e^{- \frac {t \left (1 - \sqrt {145}\right )}{8}}\right ]
\]