60.3.2 problem Problem 3
Internal
problem
ID
[15226]
Book
:
Differential
equations
and
the
calculus
of
variations
by
L.
ElSGOLTS.
MIR
PUBLISHERS,
MOSCOW,
Third
printing
1977.
Section
:
Chapter
3,
SYSTEMS
OF
DIFFERENTIAL
EQUATIONS.
Problems
page
209
Problem
number
:
Problem
3
Date
solved
:
Thursday, October 02, 2025 at 10:07:31 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+5 x \left (t \right )+y \left (t \right )&={\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )-x \left (t \right )-3 y \left (t \right )&={\mathrm e}^{2 t} \end{align*}
✓ Maple. Time used: 0.223 (sec). Leaf size: 101
ode:=[diff(x(t),t)+5*x(t)+y(t) = exp(t), diff(y(t),t)-x(t)-3*y(t) = exp(2*t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\left (-1+\sqrt {15}\right ) t} c_2 +{\mathrm e}^{-\left (1+\sqrt {15}\right ) t} c_1 +\frac {2 \,{\mathrm e}^{t}}{11}+\frac {{\mathrm e}^{2 t}}{6} \\
y \left (t \right ) &= -{\mathrm e}^{\left (-1+\sqrt {15}\right ) t} c_2 \sqrt {15}+{\mathrm e}^{-\left (1+\sqrt {15}\right ) t} c_1 \sqrt {15}-4 \,{\mathrm e}^{\left (-1+\sqrt {15}\right ) t} c_2 -4 \,{\mathrm e}^{-\left (1+\sqrt {15}\right ) t} c_1 -\frac {7 \,{\mathrm e}^{2 t}}{6}-\frac {{\mathrm e}^{t}}{11} \\
\end{align*}
✓ Mathematica. Time used: 1.258 (sec). Leaf size: 206
ode={D[x[t],t]+5*x[t]+y[t]==Exp[t],D[y[t],t]-x[t]-3*y[t]==Exp[2*t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{330} e^{-\left (\left (1+\sqrt {15}\right ) t\right )} \left (60 e^{\left (2+\sqrt {15}\right ) t}+55 e^{\left (3+\sqrt {15}\right ) t}-11 \left (\left (4 \sqrt {15}-15\right ) c_1+\sqrt {15} c_2\right ) e^{2 \sqrt {15} t}+11 \left (\left (15+4 \sqrt {15}\right ) c_1+\sqrt {15} c_2\right )\right )\\ y(t)&\to -\frac {1}{330} e^{-\left (\left (1+\sqrt {15}\right ) t\right )} \left (30 e^{\left (2+\sqrt {15}\right ) t}+385 e^{\left (3+\sqrt {15}\right ) t}-11 \left (\sqrt {15} c_1+\left (15+4 \sqrt {15}\right ) c_2\right ) e^{2 \sqrt {15} t}+11 \left (\sqrt {15} c_1+\left (4 \sqrt {15}-15\right ) c_2\right )\right ) \end{align*}
✓ Sympy. Time used: 0.776 (sec). Leaf size: 92
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(5*x(t) + y(t) - exp(t) + Derivative(x(t), t),0),Eq(-x(t) - 3*y(t) - exp(2*t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - C_{1} \left (\sqrt {15} + 4\right ) e^{- t \left (1 + \sqrt {15}\right )} - C_{2} \left (4 - \sqrt {15}\right ) e^{- t \left (1 - \sqrt {15}\right )} + \frac {e^{2 t}}{6} + \frac {2 e^{t}}{11}, \ y{\left (t \right )} = C_{1} e^{- t \left (1 + \sqrt {15}\right )} + C_{2} e^{- t \left (1 - \sqrt {15}\right )} - \frac {7 e^{2 t}}{6} - \frac {e^{t}}{11}\right ]
\]