62.39.1 problem Ex 1
Internal
problem
ID
[12953]
Book
:
An
elementary
treatise
on
differential
equations
by
Abraham
Cohen.
DC
heath
publishers.
1906
Section
:
Chapter
X,
System
of
simulataneous
equations.
Article
64.
Systems
of
linear
equations
with
constant
coefficients.
Page
150
Problem
number
:
Ex
1
Date
solved
:
Monday, March 31, 2025 at 07:27:59 AM
CAS
classification
:
system_of_ODEs
\begin{align*} 3 \frac {d}{d t}x \left (t \right )+3 x \left (t \right )+2 y \left (t \right )&={\mathrm e}^{t}\\ 4 x \left (t \right )-3 \frac {d}{d t}y \left (t \right )+3 y \left (t \right )&=3 t \end{align*}
✓ Maple. Time used: 0.183 (sec). Leaf size: 46
ode:=[3*diff(x(t),t)+3*x(t)+2*y(t) = exp(t), 4*x(t)-3*diff(y(t),t)+3*y(t) = 3*t];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{3}} c_2}{2}-{\mathrm e}^{-\frac {t}{3}} c_1 -6 t \\
y \left (t \right ) &= {\mathrm e}^{\frac {t}{3}} c_2 +{\mathrm e}^{-\frac {t}{3}} c_1 +9 t +9+\frac {{\mathrm e}^{t}}{2} \\
\end{align*}
✓ Mathematica. Time used: 1.071 (sec). Leaf size: 337
ode={3*D[x[t],t]+3*x[t]+2*y[t]==Exp[t],4*x[t]-3*D[y[t],t]+3*y[t]==3*t};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to e^{-t/3} \left (-\left (e^{2 t/3}-2\right ) \int _1^t\frac {1}{3} e^{-\frac {K[1]}{3}} \left (-3 e^{\frac {2 K[1]}{3}} K[1]+3 K[1]-e^{K[1]}+2 e^{\frac {5 K[1]}{3}}\right )dK[1]-\left (e^{2 t/3}-1\right ) \int _1^t\frac {1}{3} e^{-\frac {K[2]}{3}} \left (3 e^{\frac {2 K[2]}{3}} K[2]-6 K[2]+2 e^{K[2]}-2 e^{\frac {5 K[2]}{3}}\right )dK[2]+c_1 \left (-e^{2 t/3}\right )-c_2 e^{2 t/3}+2 c_1+c_2\right ) \\
y(t)\to e^{-t/3} \left (2 \left (e^{2 t/3}-1\right ) \int _1^t\frac {1}{3} e^{-\frac {K[1]}{3}} \left (-3 e^{\frac {2 K[1]}{3}} K[1]+3 K[1]-e^{K[1]}+2 e^{\frac {5 K[1]}{3}}\right )dK[1]+\left (2 e^{2 t/3}-1\right ) \int _1^t\frac {1}{3} e^{-\frac {K[2]}{3}} \left (3 e^{\frac {2 K[2]}{3}} K[2]-6 K[2]+2 e^{K[2]}-2 e^{\frac {5 K[2]}{3}}\right )dK[2]+2 c_1 e^{2 t/3}+2 c_2 e^{2 t/3}-2 c_1-c_2\right ) \\
\end{align*}
✓ Sympy. Time used: 0.227 (sec). Leaf size: 48
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(3*x(t) + 2*y(t) - exp(t) + 3*Derivative(x(t), t),0),Eq(-3*t + 4*x(t) + 3*y(t) - 3*Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - C_{1} e^{- \frac {t}{3}} - \frac {C_{2} e^{\frac {t}{3}}}{2} - 6 t, \ y{\left (t \right )} = C_{1} e^{- \frac {t}{3}} + C_{2} e^{\frac {t}{3}} + 9 t + \frac {e^{t}}{2} + 9\right ]
\]