75.27.5 problem 780
Internal
problem
ID
[17162]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Chapter
3
(Systems
of
differential
equations).
Section
20.
The
method
of
elimination.
Exercises
page
212
Problem
number
:
780
Date
solved
:
Thursday, March 13, 2025 at 09:18:06 AM
CAS
classification
:
system_of_ODEs
\begin{align*} 4 \frac {d}{d t}x \left (t \right )-\frac {d}{d t}y \left (t \right )+3 x \left (t \right )&=\sin \left (t \right )\\ \frac {d}{d t}x \left (t \right )+y \left (t \right )&=\cos \left (t \right ) \end{align*}
✓ Maple. Time used: 0.056 (sec). Leaf size: 36
ode:=[4*diff(x(t),t)-diff(y(t),t)+3*x(t) = sin(t), diff(x(t),t)+y(t) = cos(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \frac {{\mathrm e}^{-3 t} c_{2}}{3}+{\mathrm e}^{-t} c_{1} \\
y \left (t \right ) &= {\mathrm e}^{-3 t} c_{2} +{\mathrm e}^{-t} c_{1} +\cos \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.056 (sec). Leaf size: 289
ode={4*D[x[t],t]-D[y[t],t]+3*x[t]==Sin[t],D[x[t],t]+y[t]==Cos[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{2} e^{-3 t} \left (\left (3 e^{2 t}-1\right ) \int _1^t\frac {1}{2} e^{K[1]} \left (\left (-1+3 e^{2 K[1]}\right ) \cos (K[1])-\left (-1+e^{2 K[1]}\right ) \sin (K[1])\right )dK[1]-\left (e^{2 t}-1\right ) \int _1^t\frac {1}{2} e^{K[2]} \left (\left (-1+9 e^{2 K[2]}\right ) \cos (K[2])+\left (1-3 e^{2 K[2]}\right ) \sin (K[2])\right )dK[2]+3 c_1 e^{2 t}-c_2 e^{2 t}-c_1+c_2\right ) \\
y(t)\to \frac {1}{2} e^{-3 t} \left (3 \left (e^{2 t}-1\right ) \int _1^t\frac {1}{2} e^{K[1]} \left (\left (-1+3 e^{2 K[1]}\right ) \cos (K[1])-\left (-1+e^{2 K[1]}\right ) \sin (K[1])\right )dK[1]-\left (e^{2 t}-3\right ) \int _1^t\frac {1}{2} e^{K[2]} \left (\left (-1+9 e^{2 K[2]}\right ) \cos (K[2])+\left (1-3 e^{2 K[2]}\right ) \sin (K[2])\right )dK[2]+3 c_1 e^{2 t}-c_2 e^{2 t}-3 c_1+3 c_2\right ) \\
\end{align*}
✓ Sympy. Time used: 0.237 (sec). Leaf size: 32
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(3*x(t) - sin(t) + 4*Derivative(x(t), t) - Derivative(y(t), t),0),Eq(y(t) - cos(t) + Derivative(x(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} e^{- t} + \frac {C_{2} e^{- 3 t}}{3}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{- 3 t} + \cos {\left (t \right )}\right ]
\]