64.16.8 problem 8
Internal
problem
ID
[13540]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
7,
Systems
of
linear
differential
equations.
Section
7.1.
Exercises
page
277
Problem
number
:
8
Date
solved
:
Monday, March 31, 2025 at 08:00:48 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-x \left (t \right )-3 y \left (t \right )&=3 t\\ \frac {d}{d t}x \left (t \right )+2 \frac {d}{d t}y \left (t \right )-2 x \left (t \right )-3 y \left (t \right )&=1 \end{align*}
✓ Maple. Time used: 0.132 (sec). Leaf size: 59
ode:=[diff(x(t),t)+diff(y(t),t)-x(t)-3*y(t) = 3*t, diff(x(t),t)+2*diff(y(t),t)-2*x(t)-3*y(t) = 1];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\sqrt {3}\, t} c_2 +{\mathrm e}^{-\sqrt {3}\, t} c_1 +3 t -3 \\
y \left (t \right ) &= \frac {\sqrt {3}\, {\mathrm e}^{\sqrt {3}\, t} c_2}{3}-\frac {\sqrt {3}\, {\mathrm e}^{-\sqrt {3}\, t} c_1}{3}+\frac {4}{3}-2 t \\
\end{align*}
✓ Mathematica. Time used: 4.596 (sec). Leaf size: 482
ode={D[x[t],t]+D[y[t],t]-x[t]-3*y[t]==3*t,D[x[t],t]+2*D[y[t],t]-2*x[t]-3*y[t]==1};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{2} e^{-\sqrt {3} t} \left (\left (e^{2 \sqrt {3} t}+1\right ) \int _1^t\frac {1}{2} e^{-\sqrt {3} K[1]} \left (-3 \left (-2+\sqrt {3}\right ) K[1]+e^{2 \sqrt {3} K[1]} \left (3 \left (2+\sqrt {3}\right ) K[1]-\sqrt {3}-1\right )+\sqrt {3}-1\right )dK[1]+\sqrt {3} \left (e^{2 \sqrt {3} t}-1\right ) \int _1^t\frac {1}{6} e^{-\sqrt {3} K[2]} \left (\left (-9+6 \sqrt {3}\right ) K[2]+e^{2 \sqrt {3} K[2]} \left (-3 \left (3+2 \sqrt {3}\right ) K[2]+\sqrt {3}+3\right )-\sqrt {3}+3\right )dK[2]+c_1 e^{2 \sqrt {3} t}+\sqrt {3} c_2 e^{2 \sqrt {3} t}+c_1-\sqrt {3} c_2\right ) \\
y(t)\to \frac {1}{6} e^{-\sqrt {3} t} \left (\sqrt {3} \left (e^{2 \sqrt {3} t}-1\right ) \int _1^t\frac {1}{2} e^{-\sqrt {3} K[1]} \left (-3 \left (-2+\sqrt {3}\right ) K[1]+e^{2 \sqrt {3} K[1]} \left (3 \left (2+\sqrt {3}\right ) K[1]-\sqrt {3}-1\right )+\sqrt {3}-1\right )dK[1]+3 \left (e^{2 \sqrt {3} t}+1\right ) \int _1^t\frac {1}{6} e^{-\sqrt {3} K[2]} \left (\left (-9+6 \sqrt {3}\right ) K[2]+e^{2 \sqrt {3} K[2]} \left (-3 \left (3+2 \sqrt {3}\right ) K[2]+\sqrt {3}+3\right )-\sqrt {3}+3\right )dK[2]+\sqrt {3} c_1 e^{2 \sqrt {3} t}+3 c_2 e^{2 \sqrt {3} t}-\sqrt {3} c_1+3 c_2\right ) \\
\end{align*}
✓ Sympy. Time used: 0.427 (sec). Leaf size: 66
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-3*t - x(t) - 3*y(t) + Derivative(x(t), t) + Derivative(y(t), t),0),Eq(-2*x(t) - 3*y(t) + Derivative(x(t), t) + 2*Derivative(y(t), t) - 1,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \sqrt {3} C_{1} e^{\sqrt {3} t} - \sqrt {3} C_{2} e^{- \sqrt {3} t} + 3 t - 3, \ y{\left (t \right )} = C_{1} e^{\sqrt {3} t} + C_{2} e^{- \sqrt {3} t} - 2 t + \frac {4}{3}\right ]
\]