71.1.159 problem 186 (page 297)
Internal
problem
ID
[19335]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
186
(page
297)
Date
solved
:
Thursday, October 02, 2025 at 04:19:00 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+5 x \left (t \right )+y \left (t \right )&=7 \,{\mathrm e}^{t}-27\\ -2 x \left (t \right )+\frac {d}{d t}y \left (t \right )+3 y \left (t \right )&=-3 \,{\mathrm e}^{t}+12 \end{align*}
✓ Maple. Time used: 0.241 (sec). Leaf size: 70
ode:=[diff(x(t),t)+5*x(t)+y(t) = 7*exp(t)-27, diff(y(t),t)-2*x(t)+3*y(t) = -3*exp(t)+12];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{-4 t} \sin \left (t \right ) c_2 +{\mathrm e}^{-4 t} \cos \left (t \right ) c_1 -\frac {93}{17}+\frac {31 \,{\mathrm e}^{t}}{26} \\
y \left (t \right ) &= -{\mathrm e}^{-4 t} \sin \left (t \right ) c_2 -{\mathrm e}^{-4 t} \cos \left (t \right ) c_2 -{\mathrm e}^{-4 t} \cos \left (t \right ) c_1 +{\mathrm e}^{-4 t} \sin \left (t \right ) c_1 -\frac {2 \,{\mathrm e}^{t}}{13}+\frac {6}{17} \\
\end{align*}
✓ Mathematica. Time used: 0.14 (sec). Leaf size: 79
ode={D[x[t],t]+5*x[t]+y[t]==7*Exp[t]-27,D[y[t],t]-2*x[t]+3*y[t]==3*Exp[t]+12};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {25 e^t}{26}+c_1 e^{-4 t} \cos (t)-(c_1+c_2) e^{-4 t} \sin (t)-\frac {93}{17}\\ y(t)&\to \frac {16 e^t}{13}+c_2 e^{-4 t} \cos (t)+(2 c_1+c_2) e^{-4 t} \sin (t)+\frac {6}{17} \end{align*}
✓ Sympy. Time used: 2.814 (sec). Leaf size: 136
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(5*x(t) + y(t) - 7*exp(t) + Derivative(x(t), t) + 27,0),Eq(-2*x(t) + 3*y(t) + 3*exp(t) + Derivative(y(t), t) - 12,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- 4 t} \sin {\left (t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- 4 t} \cos {\left (t \right )} + \frac {31 e^{t} \sin ^{2}{\left (t \right )}}{26} + \frac {31 e^{t} \cos ^{2}{\left (t \right )}}{26} - \frac {93 \sin ^{2}{\left (t \right )}}{17} - \frac {93 \cos ^{2}{\left (t \right )}}{17}, \ y{\left (t \right )} = C_{1} e^{- 4 t} \cos {\left (t \right )} - C_{2} e^{- 4 t} \sin {\left (t \right )} - \frac {2 e^{t} \sin ^{2}{\left (t \right )}}{13} - \frac {2 e^{t} \cos ^{2}{\left (t \right )}}{13} + \frac {6 \sin ^{2}{\left (t \right )}}{17} + \frac {6 \cos ^{2}{\left (t \right )}}{17}\right ]
\]