60.9.16 problem 1871
Internal
problem
ID
[11795]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1871
Date
solved
:
Wednesday, March 05, 2025 at 03:07:08 PM
CAS
classification
:
system_of_ODEs
\begin{align*} 4 \frac {d}{d t}x \left (t \right )+9 \frac {d}{d t}y \left (t \right )+2 x \left (t \right )+31 y \left (t \right )&={\mathrm e}^{t}\\ 3 \frac {d}{d t}x \left (t \right )+7 \frac {d}{d t}y \left (t \right )+x \left (t \right )+24 y \left (t \right )&=3 \end{align*}
✓ Maple. Time used: 0.191 (sec). Leaf size: 70
ode:=[4*diff(x(t),t)+9*diff(y(t),t)+2*x(t)+31*y(t) = exp(t), 3*diff(x(t),t)+7*diff(y(t),t)+x(t)+24*y(t) = 3];
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: 0.19 (sec). Leaf size: 233
ode={4*D[x[t],t]+9*D[y[t],t]+2*x[t]+31*y[t]==Exp[t],3*D[x[t],t]+7*D[y[t],t]+x[t]+24*y[t]==3};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to e^{-4 t} \left ((\cos (t)-\sin (t)) \int _1^te^{4 K[1]} \left (\left (-27+7 e^{K[1]}\right ) \cos (K[1])+\left (-15+4 e^{K[1]}\right ) \sin (K[1])\right )dK[1]-\sin (t) \int _1^t-e^{4 K[2]} \left (3 \left (-4+e^{K[2]}\right ) \cos (K[2])+\left (-42+11 e^{K[2]}\right ) \sin (K[2])\right )dK[2]+c_1 \cos (t)-c_1 \sin (t)-c_2 \sin (t)\right ) \\
y(t)\to e^{-4 t} \left (2 \sin (t) \int _1^te^{4 K[1]} \left (\left (-27+7 e^{K[1]}\right ) \cos (K[1])+\left (-15+4 e^{K[1]}\right ) \sin (K[1])\right )dK[1]+(\sin (t)+\cos (t)) \int _1^t-e^{4 K[2]} \left (3 \left (-4+e^{K[2]}\right ) \cos (K[2])+\left (-42+11 e^{K[2]}\right ) \sin (K[2])\right )dK[2]+c_2 \cos (t)+2 c_1 \sin (t)+c_2 \sin (t)\right ) \\
\end{align*}
✓ Sympy. Time used: 3.884 (sec). Leaf size: 136
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(2*x(t) + 31*y(t) - exp(t) + 4*Derivative(x(t), t) + 9*Derivative(y(t), t),0),Eq(x(t) + 24*y(t) + 3*Derivative(x(t), t) + 7*Derivative(y(t), t) - 3,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- 4 t} \sin {\left (t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- 4 t} \cos {\left (t \right )} + \frac {31 e^{t} \sin ^{2}{\left (t \right )}}{26} + \frac {31 e^{t} \cos ^{2}{\left (t \right )}}{26} - \frac {93 \sin ^{2}{\left (t \right )}}{17} - \frac {93 \cos ^{2}{\left (t \right )}}{17}, \ y{\left (t \right )} = C_{1} e^{- 4 t} \cos {\left (t \right )} - C_{2} e^{- 4 t} \sin {\left (t \right )} - \frac {2 e^{t} \sin ^{2}{\left (t \right )}}{13} - \frac {2 e^{t} \cos ^{2}{\left (t \right )}}{13} + \frac {6 \sin ^{2}{\left (t \right )}}{17} + \frac {6 \cos ^{2}{\left (t \right )}}{17}\right ]
\]