75.31.9 problem 823
Internal
problem
ID
[17197]
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
23.2
The
method
of
undetermined
coefficients.
Exercises
page
239
Problem
number
:
823
Date
solved
:
Thursday, March 13, 2025 at 09:18:39 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+y \left (t \right )-2 z \left (t \right )+2-t\\ \frac {d}{d t}y \left (t \right )&=1-x \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )+y \left (t \right )-z \left (t \right )+1-t \end{align*}
✓ Maple. Time used: 0.080 (sec). Leaf size: 50
ode:=[diff(x(t),t) = 2*x(t)+y(t)-2*z(t)+2-t, diff(y(t),t) = -x(t)+1, diff(z(t),t) = x(t)+y(t)-z(t)+1-t];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{1} \sin \left (t \right )-c_{2} {\mathrm e}^{t}-\cos \left (t \right ) c_{3} \\
y \left (t \right ) &= t +\cos \left (t \right ) c_{1} +c_{2} {\mathrm e}^{t}+\sin \left (t \right ) c_{3} \\
z &= 1+c_{1} \sin \left (t \right )-\cos \left (t \right ) c_{3} \\
\end{align*}
✓ Mathematica. Time used: 0.21 (sec). Leaf size: 401
ode={D[x[t],t]==2*x[t]+y[t]-2*z[t]+2-t,D[y[t],t]==1-x[t],D[z[t],t]==x[t]+y[t]-z[t]+1-t};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \sin (t) \int _1^t\left (2 \cos (K[2])-e^{-K[2]}-K[2] \sin (K[2])+\sin (K[2])\right )dK[2]+\left (e^t+\sin (t)\right ) \int _1^t\left (-K[1] \cos (K[1])+\cos (K[1])+e^{-K[1]}-2 \sin (K[1])\right )dK[1]-\left (e^t+\sin (t)-\cos (t)\right ) \int _1^t(-K[3] \cos (K[3])+\cos (K[3])-2 \sin (K[3]))dK[3]+c_2 \sin (t)+c_1 \left (e^t+\sin (t)\right )-c_3 \left (e^t+\sin (t)-\cos (t)\right ) \\
y(t)\to \cos (t) \int _1^t\left (2 \cos (K[2])-e^{-K[2]}-K[2] \sin (K[2])+\sin (K[2])\right )dK[2]+\left (\cos (t)-e^t\right ) \int _1^t\left (-K[1] \cos (K[1])+\cos (K[1])+e^{-K[1]}-2 \sin (K[1])\right )dK[1]-\left (-e^t+\sin (t)+\cos (t)\right ) \int _1^t(-K[3] \cos (K[3])+\cos (K[3])-2 \sin (K[3]))dK[3]+c_2 \cos (t)+c_1 \left (\cos (t)-e^t\right )-c_3 \left (-e^t+\sin (t)+\cos (t)\right ) \\
z(t)\to (\cos (t)-\sin (t)) \int _1^t(-K[3] \cos (K[3])+\cos (K[3])-2 \sin (K[3]))dK[3]+\sin (t) \int _1^t\left (-K[1] \cos (K[1])+\cos (K[1])+e^{-K[1]}-2 \sin (K[1])\right )dK[1]+\sin (t) \int _1^t\left (2 \cos (K[2])-e^{-K[2]}-K[2] \sin (K[2])+\sin (K[2])\right )dK[2]+c_1 \sin (t)+c_2 \sin (t)+c_3 (\cos (t)-\sin (t)) \\
\end{align*}
✓ Sympy. Time used: 0.250 (sec). Leaf size: 94
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(t - 2*x(t) - y(t) + 2*z(t) + Derivative(x(t), t) - 2,0),Eq(x(t) + Derivative(y(t), t) - 1,0),Eq(t - x(t) - y(t) + z(t) + Derivative(z(t), t) - 1,0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - C_{1} e^{t} - C_{2} \sin {\left (t \right )} + C_{3} \cos {\left (t \right )} + \sin ^{2}{\left (t \right )} + \cos ^{2}{\left (t \right )} - 1, \ y{\left (t \right )} = C_{1} e^{t} - C_{2} \cos {\left (t \right )} - C_{3} \sin {\left (t \right )} + t \sin ^{2}{\left (t \right )} + t \cos ^{2}{\left (t \right )} - \sin ^{2}{\left (t \right )} - \cos ^{2}{\left (t \right )} + 1, \ z{\left (t \right )} = - C_{2} \sin {\left (t \right )} + C_{3} \cos {\left (t \right )} + \sin ^{2}{\left (t \right )} + \cos ^{2}{\left (t \right )}\right ]
\]